
Class ~TJz 6^ 



Book 



Copight]^" 



COPVRIGHT DEPOSIT 



american ^eclianical CDngineering §)erie0 

MORTIMER E. COOLEY 

GENERAL EDITOR 



PRINCIPLES 



OF 



THERMODYNAMICS 



BY 



G. A. GOODENOUGH, M.E. 

M 
PROFESSOR OF THERMODYNAMICS IN THE UNIVERSITY 
OF ILLINOIS 




NEW YORK 
HENRY HOLT AND COMPANY 

1911 



/^- 



Copyright, 1911 

BY 

HENRY HOLT AND COMPANY 



.y II- 



^'cuaosfis? 




PREFACE 

This book is intended primarily for students of engineering. 
Its purpose is to provide a course in the principles of thermo- 
dynamics that may serve as an adequate foundation for the 
advanced study of heat engines. As indicated by the title, 
emphasis is placed on the principles rather than on the appli- 
cations of thermodynamics. In the chapters on the technical 
applications the underlying theory of various heat engines is 
quite fully developed. The discussion, however, is restricted 
to ideal cases, and questions that involve the design, operation, 
or performance of heat engines are reserved for a second 
volume. 

The arrangement of the subject matter and the method of 
presentation are the result of some twelve years' experience in 
teaching thermodynamics. Briefly, the arrangement is as fol- 
lows : In the first six chapters, the fundamental laws are 
developed and the general equations of thermodynamics are 
derived. The laws of gases and gaseous mixtures are dis- 
cussed in Chapters VII and VIII, and this discussion is fol- 
lowed immediately by the technical applications in which 
gaseous media play a part. A discussion of the properties 
of saturated and superheated vapors is likewise followed by the 
technical applications that involve vapor media. 

Some of the features of the book to which attention may 
be directed are the following : 

1. The method of presenting the fundamental laws. In 
this treatment I have followed very closely the development 
in Bryan's thermodynamics. The second law is made identical 
with the law of degradation of energy, the connection between 
irreversibility and loss of availability is pointed out, and by 
means of the Carnot cycle a measure of availability is obtained. 
Entropy is then defined in terms of unavailable energy, and 



iv PREFACE 

from this fundamental definition the usual definition of the 
entropy of a non-isolated system as the integral I -~ is easily 

derived. By this method of presentation, a definite concep- 
tion of the meaning and scope of the second law is obtained, 
and the difficulties that usually surround the definition of 
entropy are removed. 

2. The discussion of saturated and superheated vapors. 
The experiments in the Munich laboratory and the researches 
of Professor Marks and Dr. Davis have furnished new and 
accurate data on the thermal properties of saturated and super- 
heated steam. In Chapters X and XI a concise but fairly 
complete account of these important researches is given. Kno- 
blauch's experiments on specific volumes have been correlated 
with the experiments on specific heat by means of the Clausius 

relation [ -^ ] = — AT( -t-^S) and equations for the specific heat, 
\dp J rp \oI Jp 

entropy, energy, and heat content of superheated steam are 
thereby deduced. These results have not hitherto been pub- 
lished. 

3. The discussion of the flow of fluids and of throttling 
processes. The applications of the throttling process are so 
important from all points of view that a separate chapter is 
devoted to them. 

4. The treatment of gaseous mixtures, Chapter VIII. An 
attempt is made to present in concise form the principles and 
methods required in the accurate analysis of the internal com- 
bustion engine. 

5. The note on the interpretation of differential expressions. 
Art. 23. This important topic should be discussed fully in 
calculus, but experience shows that students rarely have a 
grasp of it. In thermodynamics the exact differential has 
extensive applications ; hence it seems desirable to include 
a rather complete explanation of exact and inexact differentials 
and their connection with thermodynamic magnitudes. A 
thorough understanding of this article should enable the student 
to pursue the subsequent mathematical discussions with intel- 
ligence and ease. 



PREFACE V 

The text is illustrated by numerous solved problems, and 
exercises are given at the ends of the chapters and elsewhere. 
Many of the exercises require only routine numerical solutions, 
but others involve the development of principles. 

References are given to the treatment of various topics in 
standard works and to original articles. It is not expected 
that undergraduate students will make extensive use of these 
references, but it is hoped that instructors and advanced 
students will find them helpful. 

In writing this book I have consulted many of the standard 
works on thermodynamics, and have made free use of whatever 
material suited my purpose. I desire to acknowledge my 
special indebtedness to the works of Brj^an, Preston, Griffiths, 
Zeuner, Chwolson, Weyrauch, and Lorenz, and to the papers 
of Dr. H. N. Davis. To Mr. John A. Dent I am indebted 
for assistance in the construction of the tables and in the 
revision of the proof sheets. Mr. A. L. Sclialler also gave 
valuable assistance in getting the book through the press. 

G. A. GOODENOUGH. 
Urbana, III., July, 1911. 



CONTENTS 

CHAPTER I 
Energy 

ART. PAGE 

1. Scope of Thermodynamics 1 

2. Energy 1 

3. Mechanical Energy 2 

4. Heat Energy .3 

5. Other Forms of Energy 5 

6. Transformations of Energy 5 

7. Conservation of Energy 6 

8. Degradation of Energy 7 

9. Units of Energy 8 

10. Units of Heat 9 

11. Relations between Energy Units 10 

CHAPTER II 
Change of State. Thermal Capacities 

12. State of a System 15 

13. Characteristic Equation 16 

14. Equation of a Perfect Gas 17 

15. Absolute Temperature 18 

16. Other Characteristic Equations 20 

17. Characteristic Surfaces 20 

18. Thermal Lines 21 

19. Heat absorbed during a Change of State 22 

20. Thermal Capacity : Specific Heat 24 

21. Latent Heat ' . 26 

22. Relations between Thermal Capacities 27 

23. Interpretation of Differential Expressions ..... 28 

CHAPTER III 

The First Law of Thermodynamics 

24. Statement of the First Law 35 

25. Effects of Heat 35 

26. Intrinsic Energy 36 

vii 



viii CONTENTS 

ART. PAGE 

27. External Work . . . . ' . . . . . .37 

28. Integration of the Energy Equation 38 

29. Energy Equation applied to a Cycle Process 39 

30. Adiabatic Processes 40 

31. Isodynamic Changes 42 

32. Graphical Representations .42 

CHAPTER IV 
The Second Law of Thermodynamics 

33. Introductory Statement 45 

34. Availability of Energy » 46 

35. Reversibility . .47 

36. General Statement of the Second Law .49 

37. Carnot's Cycle 50 

38. Carnot's Principle 52 

39. Efficiency of the Carnot Cycle 54 

40. Available Energy and Waste 56 

41. Entropy .58 

42. Second Definition of Entropy ........ 60 

43. The Inequality of Clausius 63 

44. Summary 63 

45. Boltzmann's Interpretation of the Second Law .... 65 

CHAPTER V 
Temperature Entropy Representation 

46. Entropy as a Coordinate 68 

47. Isothermals and Adiabatics 69 

48. The Curve of Heating and Cooling . . . . . .70 

49. Cycle Processes 72 

50. The Rectangular Cycle 73 

51. Internal Frictional Processes . . . . . . . .74 

52. Cycles with Irreversible Adiabatics 75 

53. Heat Content 76 

CHAPTER VI 
General Equations of Thermodynamics 

54. Fundamental Differentials 79 

55. The Thermodynamic Relations 80 

56. General Differential Equations 82 

57. Additional Thermodynamic Formulas 84 

58. Equilibrium 87 



CONTENTS ix 

4 

CHAPTER Vir 
Properties of Gases 

ART. PAGE 

59. The Permanent Gases 89 

60. Experimental Laws .89 

61. Comparison of Temperature Scales . 91 

62. Numerical Value of B 92 

63. Forms of the Characteristic Equation -.93 

64. General Equations for Gases 94 

65. Specific Heat of Gases 96 

66. Intrinsic Energy 97 

67. Heat Content 99 

68. Entropy of Permanent Gases 100 

69. Constant Volume and Constant Pressure Changes .... 101 

70. Isothermal Change of State 102 

71. Adiabatic Change of State 102 

72. Polytropic Change of State 104 

73. Specific Heat in Polytropic Changes . . . . . . 106 

74. Determination of n 108 

CHAPTER VIII 
Gaseous Compounds and Mixtures. Combustion 

75. Preliminary Statement Ill 

76. Atomic and Molecular Weights . Ill 

77. Relations between Gas Constants 112 

78. Mixtures of Gases. Dalton's Law . . . . . . . 114 

79. Volume Relations 116 

80. Combustion: Fuels 117 

81. Air required for Combustion. Products of Combustion . . 119 

82. Specific Heat of Gaseous Products 123 

83. Specific Heat of a Gaseous Mixture 125 

84. Adiabatic Changes with Varying Specific Heats .... 126 

85. Temperature of Combustion 127 

CHAPTER IX 

Technical Applications. Gaseous Media 

86. Cycle Processes . . 133 

87. The Carnot Cycle 134 

88. Conditions of Maximum Efficiency . 135 

89. Isoadiabatic Cycles 136 

90. Classification of Air Engines 137 

91. Stirling's Engine 138 

92. Ericsson's Engine . . . ' . . ' . . . . . 139 



X CONTENTS 

ART. PACK 

93. Analysis of Cycles . . 140 

94. Heating by Internal Combustion 141 

95. The Otto Cycle 142 

96. The Joule, or Brayton, Cycle 145 

97. The Diesel Cycle . . . . 146 

98. Comparison of Cycles 148 

99. Closer Analysis of the Otto Cycle .148 

100. Air Refrigeration 149 

101. Air Compression 152 

102. Water Jacketing . 155 

103. Compound Compression . . . . . , . .156 

104. Compressed-air Engines 158 

105. T^AS-Diagram of Combined Compressor and Engine . . .158 



CHAPTER X 
Saturated Vapors 

106. The Process of Vaporization .... 

107. Functional Relations. Characteristic Surfaces 

108. Relation between Pressure and Temperature 



109. Expression for 



dp 
dt 



110. Energy Equation applied to Vaporization 

111. Heat Content of a Saturated Vapor 

112. Thermal Properties of Water Vapor 

113. Heat of the Liquid .... 

114. Latent Heat of Vaporization . 

115. Total Heat. Heat Content . 

116. Specific Volume of Steam 

117. Entropy of Liquid and of Vapor . 

118. Steam Tables 

119. Properties of Saturated Ammonia . 

120. Other Saturated Vapors . 

121. Liquid and Saturation Curves 

122. Specific Heat of a Saturated Vapor 

123. General Equation for Vapor Mixtures 

124. Variation of x during Adiabatic Changes 

125. Special Curves on the T/S-plane 

126. Special Changes of State 

127. Approximate Equation for the Adiabatic of a Vapor Mixture 



164 

166 
167 

170 

170 
173 
173 
174 
175 
177 
177 
179 
180 
180 
181 
182 
182 
184 
185 
186 
188 
190 



CHAPTER XT 
Superheated Vapors 

128. General Characteristics of Superheated Vapors 

129. Critical States 



196 
197 



CONTENTS 



XI 



ART. 

130. Equations of van der Waals and Clausius < 

131. Experiments of Knoblauch, Linde, and Klebe 

132. Equations for Superheated Steam . 

133. Specific Heat of Superheated Steam 

134. Mean Specific Heat 

135. Heat Content. Total Heat 

136. Intrinsic Energy 

137. Entropy .... 

138. Special Changes of State 

139. Approximate Equations for Adiabatic Changes 

140. Tables and Diagrams for Superheated Steam 

141. Superheated Ammonia and Sulphur Dioxide 



PAGE 

200 
201 
203 
204 
210 
210 
214 
215 
216 
220 
221 
223 



CHAPTER XII 
Mixtures of Gases and Vapors 

142. Moisture in the Atmosphere . 

143. Constants for Moist Air . 

144. Mixture of Wet Steam and Air 

145. Isothermal Change of State . 

146. Adiabatic Change of State 

147. Mixture of Air with High-pressure Steam 



228 
230 
232 
232 
233 
236 



CHAPTER XIII 
The Flow of Fluids 

148. Preliminary Statement 243 

149. Assumptions 244 

150. Fundamental Equations 244 

151. Special Forms of the Fundamental Equation .... 247 

152. Graphical Representation . . . . . . . . 247 

153. Flow through Orifices. Saint Venant's Hypothesis . . . 252 

154. Formulas for Discharge 255 

155. Acoustic Velocity . . . . 2.57 

156. The de Laval Nozzle 258 

157. Friction in Nozzles 262 

158. Desiorn of Nozzles . . 264 



CHAPTER XIV 
Throttling Processes 

159. Wiredrawing . .268 

160. Loss due to Throttling 269 

161. The Throttling Calorimeter 271 



xii CONTENTS 

ART. PAGE 

162. The Expansion Valve . . . . . . . . .272 

163. Throttling Curves . .273 

164. The Davis Formula for Heat Content 274 

165. The Joule-Thomson Effect 275 

166. Characteristic Equation of Permanent Gases .... 277 

167. Linde's Process for the Liquefaction of G-ases ...» 280 

CHAPTER XV 
Technical Applications, Vapor Media 

The Steam Engine 



168. The Carnot Cycle for Saturated Vapors 

169. The Rankine Cycle 

170. The Rankine Cycle with Superheated Steam 

171. Incomplete Expansion 

172. Effect of Changing the Limiting Pressures . 

173. Imperfections of the Actual Cycle . 

174. Efficiency Standards 



283 
284 
286 
288 
289 
290 
291 



The Steam Turbine 



175. Comparison of the Steam Turbine and Reciprocating Engine . 294 

176. Classification of Steam Turbines 295 

177. Compounding 296 

178. Work of a Jet 298 

179. Single-stage Velocity Turbine 300 

180. Multiple-stage Velocity Turbine 302 

181. Turbine with both Pressure and Velocity Stages .... 304 

182. Pressure Turbine 305 

183. Influence of High Vacuum .307 

Refrigeration with Vapor Media 

184. Compression Refrigerating Machines 308 

185. Vapors used in Refrigeration 310 

186. Analysis of a Vapor Machine 311 



SYMBOLS 

Note. The following list gives the symbols used in this book. In case 
a magnitude is dependent upon the weight of the substance, the small 
letter denotes the magnitude referred to unit weight, the capital letter the 
same magnitude referred to M units of weight. Thus q denotes the heat 

absorbed by one pound of a substance, Q = Mq, the heat absorbed by M 
pounds. 

./, Joule's equivalent. 

.4, reciprocal of Joule's equivalent. 

3/, weight of system under consideration. 

t, temperature on the F. or the C. scale. 

T, absolute temperature. 

p, pressure. 

V, V, volume. 

y, specific weight ; also heat capacity. 

</, U, intrinsic energy of a system. 

i, I, heat content at constant pressure. 

s, S, entropy. 

W, external work. 

q, Q, heat absorbed by a system from external sources. 

h, H, heat generated within a system by irreversible transformation of 
work into heat. 

c, specific heat. 

c^,, specific heat at constant volume. 

Cp, specific heat at constant pressure. 

k, ratio Cp/cv 

B, constant in the gas equation po = BT. 

R, universal gas constant. 

n, exponent in equation for polytropic change, pV* = C. 

m, molecular weight. 

«i, a2.-, atomic weights. 

Hj„, heating value of a fuel mixture. 

X, quality of a vapor mixture (p. 165). 

q', heat of the liquid. 

q", total heat of saturated vapor, 

r, latent heat of vaporization, 

p, internal latent heat. 

ip, external latent heat. 



xiv SYMBOLS 



ir , V 



specific volume of liquid and of vapor, respectively, 
w", u", internal energy of liquid and of vapor, respectively. 
s', s", entropy of liquid and of vapor, respectively. 
i', i", heat content of liquid and of vapor, respectively. 
c', c", specific heat of liquid and of vapor, respectively. 
<^, humidity. 

velocity of flow, 
acoustic velocity. 
jP, area of cross-section of channel. 
3, work of overcoming friction in the flow of fluids. 
Pm, critical pressure (flow of fluids). 
fx, Joule-Thomson coefficient. 
7], efficiency of a heat engine. 
N, steam consumption per h.p.-hour. 






PRmCIPLES OF THERMODYNAMICS 



CHAPTER I 

ENERGY 

1. Scope of Thermodynamics. — In the most general sense, 
thermodynamics is the science that deals with energy. Since 
all natural phenomena, all physical processes, involve manifes- 
tations of energy, it follows that thermodynamics is one of 
the most fundamental and far-reaching of sciences. Thermo- 
dynamics lies at the foundation of a large region of physics 
and also of a large region of chemistry ; and it stands in a 
more or less intimate relation with other sciences. 

In a more restricted sense, thermodynamics is that branch of 
physics which deals specially with a form of energy called heat. 
It deals with transformations of heat energy into other forms 
of energy, develops the laws that govern such transformations, 
and investigates the properties of the media by which the 
transformations are effected. In technical thermodynamics the 
general principles thus developed are applied to the problems 
presented by the various heat motors. 

In this volume the general principles of thermodynamics are 
developed so far as is essential to give a firm foundation for the 
technical applications in engineering practice. The scope of 
the book does not permit a discussion of the methods of inves- 
tigation that are employed so fruitfully in physics and chem- 
istry. 

2. Energy. — A body or system of bodies is said to possess 
energy when by virtue of its condition the body or system is 
capable of doing mechanical work while undergoing a change 
of state. Many illustrations of this statement will occur to 
the reader. A moving body is capable of doing work in com- 



2 ENERGY [chap, i 

ing to rest, that is, in changing its state as regards velocity ; 
a body in an elevated position can do work in changing its 
position ; a heated metal rod is capable of doing mechanical 
work when it contracts in cooling. In each case some change 
in the state of the body results in the doing of work ; hence, in 
each case the body in question possesses energy. 

Energy, like motion, is purely relative. It is impossible to 
give a numerical value to the energy of a system without 
referring it to some standard system, whose energy we may 
arbitrarily assume to be zero. For example, the energy of the 
water in an elevated reservoir is considered with reference to 
the energy of an equal quantity at some chosen lower level. 
The kinetic energy of a body moving with a definite velocity 
is compared with that of a body at rest on the eartli's surface, 
and having, therefore, zero velocity relative to the earth. The 
energy of a pound of steam is referred to that of a pound of 
water at the temperature of melting ice. 

3. Mechanical Energy is that possessed by a body or system 
due to the motion or position of the body or system relative to 
some standard of reference. Mechanical kinetic energy is that 
due to the motion of a body and is measured by the product 
I mv^^ w^here m denotes the mass of the body and v its velocity 
relative to the reference system. It should be observed that 
\ mv^ is a scalar, not a vector, quantity and it must be considered 
positive in sign. Hence, if a system consists of a number of 
masses m^, m<^^ •••, m^ moving with velocities Vj, v^^ — , v„, 
respectively, the total kinetic energy of the system is the sum 

^ (m-^v-^ + m^v^ + • • • + infin^n) =■ 2 21 m?;^, 

independently of the directions of the several velocities. 

Tlie mechanical potential energy of a system is that due to 
position or configuration, and may be defined as the work 
the system can do in passing from its given position or con- 
figuration to a standard reference position or configuration. 
Examples of sj^stems having mechanical potential energy are 
seen in elevated reservoirs of water, stretched or compressed 
springs, etc. 



ART. 4] HEAT ENERGY 3 

4. Heat Energy. — Heat is the name given to an active agent 
postulated to account for changes in temperature. It is ob- 
served that when two bodies are placed in communication, the 
temperature of the warmer falls, that of the colder rises, and the 
change continues until the two bodies attain the same tempera- 
ture. To account for this phenomenon we say that heat flows 
from the hotter to the colder body. The fall of temperature of 
one body is due to the loss of heat, while the rise in tempera- 
ture of the other is due to the heat received by it. It is to be 
noted that the change of temperature is the thing observed and 
that the idea of heat is introduced to account, for the change, 
just as in dynamics the idea of force is introduced to account 
for the observed motion of bodies. Whatever may be the 
nature of heat, it is evidently something measurable, something 
possessing the characteristics of quantity. 

In the old caloric theory, heat was assumed to be an impon- 
derable, all-pervading fluid which could pass from one body to 
another and thus cause changes of temperature. The experi- 
ments of Rumford (1798), Davy (1812), and Joule (1840) 
shattered the caloric theory and established the modern me- 
chanical theory, of which the following is a brief outline. 

Heat may be generated by the expenditure of mechanical 
work. Familiar examples are shown in the heating of journals 
due to friction, the heating of air by compression, the develop- 
ment of heat by impact, etc. Conversely, work may be ob- 
tained by the expenditure of heat, as exemplified in the steam 
engine and other heat motors. Joule's experiments established 
the fact that a definite relation exists between the heat gener- 
ated and the work expended; thus to produce a unit of heat a 
definite amount of work is required, no matter in what particu- 
lar way the work is done. Heat and mechanical energy are 
therefore equivalent in a certain sense. Either may be produced 
at the expense of the other, and the ratio between the quantity of 
one produced and the quantity of the other expended is always 
the same. The conclusion is evident that heat is not a sub- 
stance but a form of energy ; and the mechanical theory asserts 
that this heat energy is due to the motion or configuration of 
the molecules of a body or system. 



4 ENERGY [chap, i 

Heat energy, like mechanical energy, may be either of the 
kinetic or the potential form. Denoting the mass of a mole- 
cule by m and the velocity by v, the kinetic energy of the mole- 
cule is J mv'^. In a given system the different molecules are 
moving with different velocities and in different directions ; 
nevertheless, the summation 

extended to all the molecules of the system gives the thermal 
kinetic energy of the system. If we denote by e^ the mean 
square of the velocities of the molecules, we have 

where i!f^ denotes the mass of the system. Considerations de- 
rived from the kinetic theory of gases show that the tempera- 
ture of the system is a function of c^ ; hence, since the kinetic 
energy is directly proportional to <?2, it follows that the tempera- 
ture of a system is a measure of its thermal kinetic energy. 
Whenever the temperature of a body rises, we infer that the 
kinetic energy has increased, and that the mean velocity of the 
molecules is greater than before. 

Potential thermal energy is due to the relative position of 
the molecules of the system. Tlie addition of heat to a body 
usually results in the expansion of the body. The molecules 
are moving with higher speeds than before the addition of heat, 
and on the whole they are farther apart. To separate them 
against their mutual attractions requires the expenditure of 
work ; conversely, in coming back to the original configura- 
tion the molecules will do work. Hence, the work expended 
in separating the molecules is stored in the system as potential 
energy. 

As long as the body remains in the same state of aggregation, 
the potential energy it is capable of storing is small. But if a 
bod}^ changes its state of aggregation, it may, during the pro- 
cess, store a large amount of potential energy. Consider, for 
example, the melting of ice. To change a unit weight of ice at 
melting temperature to water requires a large amount of heat. 
No part of the heat thus received by the system is stored as kinetic 
energy because during the process the temperature remains sta- 



ART. 6] TRANSFORMATIONS OF ENERGY 5 

tionary. All the heat is used in breaking down the molecular, 
structure of the solid ice and changing it to that of the liquid 
water. The heat is therefore stored as potential energy. In the 
same manner when water is transformed into steam, work is 
done in forcing apart the molecules against their cohesive forces, 
and this work is stored as potential energy. 

5. Other Forms of Energy. — In addition to heat and mechani- 
cal energy, there are other forms of energy that require consid- 
eration. The energy stored in fuel or in explosives may be 
considered potential chemical energy. Electrical energy is 
exemplified in the electric current and in the electrostatic charge 
in a condenser. Other forms of energy are due to wave motions 
either in ordinary fluid media or in the ether. Sound, for 
example, is a wave motion usually in air. Light and radiant 
heat are wave motions in the ether. 

The vibratory forms of energy are neither kinetic nor potential, 
but rather periodic alternations between the two. To illustrate 
this statement, let us consider the motion of a pendulum bob. 
In its lowest position the bob has zero potential energy and 
maximum kinetic energy ; as it rises its velocity decreases ; 
therefore, its kinetic energy also decreases, while its potential 
energy simultaneously increases and reaches a maximum at the 
end of the swing when the kinetic energy is zero. This same 
alternation from kinetic to potential and back occurs in vibrating 
strings, water waves, and, in fact, in all wave motions. 

6. Transformations of Energy. — Attention has been called 
to the generation of heat energy by the expenditure of mechani- 
cal work. This is only one of a great number of energy changes 
that are continually occurring. We see everywhere in every- 
day life one kind of energy disappearing and another form 
simultaneously appearing. In a power station, for example, 
the potential energy stored in the coal is liberated and is used up 
in adding heat energy to the water in the boiler. Part of this 
heat energy disappears in the engine and its equivalent appears 
as mechanical work. Finally, this work is expended in driving 
a generator, and in place of it appears electric energy in the 
form of the current in the circuit. We say in such cases that 



6 ENERGY , [chap, i 

one form of energy is transformed into another. The follow- 
ing are a few familiar examples of energy transformations ; 
many others will occur to the reader. 

Mechanical to heat : Compression of gases ; friction ; im- 
pact. 

Heat to mechanical : Steam engine ; expansion and contrac- 
tion of bodies. 

Mechanical to electrical : Dynamo ; electric machine. 

Electrical to mechanical : Electric motor. 

Heat to electrical : Thermopile. 

Electrical to heat : Heating of conductors by current. 

Chemical to electrical : Primary or secondary battery. 

Electrical to chemical : Electrolysis. 

Chemical to thermal : Combustion of fuel. 

7. Conservation of Energy. — Experience points to a general 
principle underlying all transformations of energy. 

The total energy of an isolated system remains constant and 
cannot he increased or diminished hy any physical processes 
whatever. 

In other words, energy, like matter, can be neither created 
nor destroyed ; whenever it apparently disappears it has been 
transformed into energy of another kind. 

This principle of the conservation of energy was first defi- 
nitely stated by Dr. J. R. Meyer in 1842, and it soon received 
confirmation from the experiments of Joule on the mechanical 
equivalent of heat. The conservation law cannot be proved 
by mathematical methods. Like other general principles in 
physics, it is founded upon experience and experiment. So 
far, it has never been contradicted by experiment, and it may 
be regarded as established as an exact law of nature. 

A perpetual Tnotion of the first class is one that would sup- 
posedly give out energy continually without any corresponding 
expenditure of energy. That is, it would create energy from 
nothing. A perpetual-motion engine would, therefore, give out 
an unlimited amount of work without fuel or other external 
supply of energy. Evidently such a machine would violate the 
conservation law ; and the statement that perpetual motion of 



ART. 8] DEGRADATION OF ENERGY 7 

the first class is impossible is equivalent to the statement of the 
conservation principle at the beginning of this article. 

8. Degradation of Energy. — While one form of energy can 
be transformed into any other form, all transformations are not 
effected with equal ease. It is only too easy to transform 
mechanical work into heat ; in fact, it is one duty of the 
engineer to prevent this transformation as far as possible. 
Furthermore, of a given amount of work all of it can be trans- 
formed into heat. The reverse transformation, on the other 
hand, is not easy of accomplishment. Heat is not transformed 
into work without effort, and of a given quantity of heat only a 
part can be thus transformed, the remainder being inevitably 
tlirown away. All other forms of energy can, like mechanical 
energy, be completely converted into heat. Electrical energy, 
for example, in the form of a current, can be thus completely 
transformed. Comparing mechanical and electrical energy, we 
see that they stand on the same footing as regards transforma- 
tion. In a perfect apparatus mechanical work can be com- 
pletely converted into electrical energy, and, conversely, electric 
energy can be completely converted into mechanical work. 

We are thus led to a classification of energy on the basis of 
the possibility of complete conversion. Energy that is capable 
of complete conversion, like mechanical and electrical energy, 
we may call high-grade energy; while heat, which is not capable 
of complete conversion, we may call low-grade energy. 

There seems to be in nature a universal tendency for energy 
to degenerate into a form less available for transformation. 
Heat will flow from a body of higher temperature to one of 
lower temperature wdth the result that a smaller fraction of it 
is available for transformation into work. High-grade energy 
tends to degenerate into loAV-grade heat energy. Thus work is 
degraded into heat through friction, and electrical energ}'' is 
rendered unavailable when transformed into heat in the con- 
ducting system. Even when one form of high-grade energy is 
transformed into another, there is an inevitable transformation 
of some part of the high-grade energy into heat. Thus in the 
electric motor a small percentage of the electric energy supplied 



8 ENERGY [chap, j 

is wasted in friction ; in the case of chemical reactions the 
chemical energy of the products is less than that of the original 
substances, the difference being due to the heat developed dur- 
ing the reaction. As Griffiths aptly says: "Each time we 
alter our investment in energy, we have thus to pay a commis- 
sion, and the tribute thus exacted can never be wholly recovered 
by us and must be regarded, not as destroyed, but as thrown on 
the waste-heap of the Universe." 

The terms degradation of energy, dissipation of energy, and 
thermodynamic degeneration are applied by different writers to 
this phenomenon that we have just described. We may for- 
mally state the principle of degradation of energy as follows : 

Every natural process is accompanied hy a certain degradation 
of energy or thermodynamic degeneration. 

The principle of the degradation of energy denies the possi- 
bility of perpetual motion of the second class, which may be de- 
scribed as follows : A mechanism with friction is inclosed in a 
case through which no energy passes. Let the mechanism be 
started in motion. Because of friction, work is converted into 
heat, which remains in the system, since no energy passes 
through the case. Suppose now that the heat thus produced 
can be transformed completely into work ; then the work may 
be used again to overcome friction and the heat thus produced 
can be again transformed into work. We then have a perpetual 
motion in a mechanism with friction without the addition of 
energy from an external source. Such a mechanism does not 
violate the conservation law, since no energy is created. It, 
however, is just as much of an absurdity as the perpetual motion 
of the first-class because it violates the principle of degradation. 

We shall discuss the degradation principle more at length in 
a subsequent chapter. 

9. Units of Energy. — According to the conservation law, 
the quantity of energy remains unchanged through all trans- 
formations. Hence, a single unit is sufficient for the measure- 
ment of energy whatever its form may be. This unit is furnished 
by the erg, the absolute unit of work in the C. G. S. system, or 
by the joule, which is 10^ ergs. It would save much confusion 



ART. 10] UNITS OF HEAT 9 

and annoyance if a single unit, as the joule, were used for all 
forms of energy. Unfortunately, however, the joule is ordina- 
rily used in connection with electrical energy only, and other 
units are used for other forms of energy. The following are 
the units generally employed. 

For mechanical energy: 

1. The foot-pound (or in the metric system, the kilogram- 

meter). This is the unit ordinarily employed by 
engineers. 

2. The horsepower-hour, which is equal to 1,980,000 foot- 

pounds. This unit is most convenient for ex- 
pressing large quantities of work. It should be 
noted that although the word " hour " is included in 
the name, the time element is in reality lacking, 
and the horsepower-hour is a unit of work, not a 
unit of power. 

For heat energy : 

1. The British thermal unit (B. t . u.). 

2. The calorie. 

The accurate definition of these thermal units and the means 
employed in establishing them demand special consideration. 

10. Units of Heat. — Obviously heat may be measured by 
observing the effects produced by it upon substances. Two of 
the most marked effects are : (1) rise of temperature ; (2) 
change of state of aggregation, as in the melting of ice or 
vaporization of water. Hence, we have two possible means of 
establishing a unit of heat : 

1. The heat required to raise a given mass of a selected 
substance, as water, through a chosen range of temperature 
may be taken as the unit. 

2. The quantity of heat required to change the state of 
aggregation of some substance, as, for example, to melt a given 
weight of ice, may be taken as the unit. 

According to Griffiths, the standard thermal unit is the heat 
required to raise 1 gram of water from 17° to 18° O. on the Paris 
hydrogen scale, or one fifth of the amount to raise it from 15° to 



10 



ENERGY 



[chap. I 



20° C on the same scale. This thermal unit is called the gram- 
calorie, or the small calorie. If the weight of water is taken as 
1 kilogram, the resulting unit is the kilogram -calorie or large 
calorie. This is the unit employed by engineers. 

The British thermal unit is defined as the heat required to 
raise the temperature of \ pound of water from 63° to 64° F. 

The method of establishing thermal units by the rise of tem- 
perature of water is open to one serious objection, namely : 
The energy required to raise the temperature of water one 
degree is quite different at different temperatures. Thus the 
number of joules required to raise a given mass of water from 

0° to 1° Cor from 99° to 
100° C. is considerably 
larger than the number 
of joules required to 
raise the same mass from 
40° to 41° C. The curve. 
Fig. 1, shows graphically 
the energy required per 
degree rise of tempera- 
ture from 0° to 100° C. 
It follows that we may 
have a number of different thermal units depending upon the 
temperature adopted in the definition. By many physicists 
the 15°-calorie is used. This is the heat required to raise the 
temperature of a gram of water from 14 J° C. to 151° C. In 
recent years there has been a tendency to unite on the so- 
called mean calorie, which may be defined as the y J^^ part of the 
heat required to raise a gram of water from 0° C. to 100° C. 
The 17-|°-calorie, as defined by Griffiths, is practically equal 
to the mean calorie. Corresponding to the mean calorie is the 
mean B. t. u., which is j|^q- of the heat required to raise the 
temperature of one pound of water from 32° to 212° F. This 
is equal to the B. t. u. at 63|-°. 



1.008 
1.006 
1.004 
1.002 
1.000 
0.998 
0,996 



■' 1 ■■■ — 


\ 


I 




V / 


3 ^/' 


V /^ 


5 / 


\ -.^ 


0° N20^ 40° 60°/ 80° 100' 


\ ^^ 


"'"T 1 — "^ 







Fig. 1. 



11. Relations between Energy Units. — The relation between 
the joule, the absolute unit of energy, and any of the gravita- 
tional units, as the foot-pound, kilogram-meter, or horsepower- 



ART. 11] RELATIONS BETWEEN ENERGY UNITS 11 

hour, is readily derived when the value of the constant g is 
given. By international agreement g is taken as 

980.665 -^ = 32.174 -^. 



sec.^ sec.^ 

The second value is obtained by means of the conversion factor 

1 cm.= 0.393T in. 
Bearing in mind the definition of the erg, we have 
1 kilogram-meter = 98066500 ergs 
= 9.80665 joules. 
Now making use of the relation 1 kg. = 2.204622 lb. and the 
preceding relation between the units of length, we readily find 
the relation 

1 foot-pound = 1.3558 joules, 
or 1 joule = 0.73756 foot-pound. 

The numerical relation between the thermal unit and the 
joule, that is, the number of joules in one gram-calorie, is called 
the mechanical equivalent and is denoted by J, The determi- 
nation of this constant has engaged the efforts of physicists 
since 1843.* 

In this work two experimental methods have been chiefly 
employed : (1) The direct method, in which mechanical energy 
is transformed directly into heat. (2) The indirect method, in 
which heat is produced by the expenditure of energy in some 
form other than mechanical. Usually electrical energy is thus 
transformed. 

The earliest experiments were those of Joule (1843), using 
the direct method. Work was expended in stirring water by 
means of a revolving paddle. From the rise of temperature 
of the known weight of water, the heat energy developed could 
be expressed in thermal units; and a comparison of this quan- 
tity with the measured quantity of work supplied gave imme- 
diately the desired value of J. 

Professor Rowland (1878-1879) used the same method, but 
by driving the paddle wheel with a petroleum engine he was 

. * For a very full description of the experiments made to determine the 
mechanical equivalent and a searching discussion of the results, the reader is 
referred to Griffiths' book, The Thermal Measurement of Energy. 



12 ENERGY [chap. I 

enabled to supply a much larger quantity of mechanical energy 
to the water, and the influence of various corrections was cor- 
respondingly decreased. Rowland's results are justly given 
great weight in deducing the finally accepted value of J. 

Another result of the highest value is that found by Rey- 
nolds and Moorby (1897). The work of a 100 horsepower 
engine was absorbed by a hydraulic brake. Water entered 
the brake at or near 0° C. and was run through it at a rate that 
caused it to emerge at a temperature of about 100° C. In this 
way the mechanical equivalent of the heat required to raise the 
temperature of one pound of water from 0° to 100° C. was 
determined. 

Of the experiments by the indirect method those of Griffiths 
(1893), Schuster and Gannon (1894), and Callendar and 
Barnes (1899) deserve mention. In each set of experiments 
the heat developed by an electric current was measured and 
compared with the electrical energy expended. 

From a careful comparison of the results of the most trust- 
worthy experiments, Griffiths has decided that the most prob- 
able value of J^is 4.184. That is, taking the 17 J-° gram-calorie, 
1 gram-calorie = 4.184 joules. 

By the use of the necessary reduction factors, we obtain the 
following relations : 

1 kg. -calorie = 426.65 kilogram-meters. 
1 B. t. u. = 777.64 foot-pounds. 

For ordinary calculations, the values 427 and 778, respectively, 
are sufficiently accurate. 

In writing some of the general equations of thermodynamics 
it is frequently convenient to use the reciprocal of J. This is 

denoted by the symbol A ; that is, A = — . We may regard A 

as the heat equivalent of work; thus 

1 ft.-lb. = J. B. t. u. 

When the horsepower-hour is taken as the unit of work, we 
have 

^1980000^2646.2. 
777.64 



ART. 11] RELATIONS BETWEEN ENERGY UNITS 13 

Hence, 1 h.p.-hr. = 2546.2 B. t. u., 

a relation that is frequently useful. 

EXERCISES 

1. If the thermal unit is taken as the heat required to raise the tempera- 
ture of 1 pound of water from 17*^ to 18° C, what is the value of / in foot- 
pounds? 

2. In the combustion of a pound of coal 13,200 B. t. u. are liberated. If 
7^ per cent of this heat is transformed into work in an engine, what is the 
coal consumption per horsepower-hour ? 

3. A gas engine is supplied with 11,200 B.t. u. per horsepower-hour. 
Find the percentage of the heat supplied that is usefully employed. 

4. In a steam engine 193 B.t.u. of the heat brought into the cylinder 
by each pound of steam is transformed into work. Find the steam con- 
sumption per horsepower-hour. 

5. The metric horsepower is defined as 75 kilogram-meters of work per 
second. Find the equivalent in kilogram-calories of a metric horsepower- 
hour. 

6. Find the numerical relations between the following energy units : 

(a) Joule and B.t. u. 

(b) Joule and naetric h.p.-hr. 

(c) B.t.u. and kg.-meter 
(^d) h.p. -minute and B.t.u. 

7. A unit of power is the watt, which is defined as 1 joule per second. 
1 kilowatt (kw.) is 1000 watts. Find the number of B. t. u. in a kw.-hr. ; 
the number of foot-pounds in a watt-hour. 

8. A Diesel oil engine may under advantageous conditions transform as 
high as 38 per cent of the heat supplied into work. If the combustion of a 
pound of oil develops 18,000 B.t. u., what weight of oil is required per h.p.-hr. ? 

REFERENCES 

The Mechanical Theory of Heat 

Rumford: Phil. Trans., 1798, 1799. 

Davy : Complete works 2, 11. 

Black : Lectures on the Elements of Chemistry 1, 33. 

Verdet: Lectures before the Chemical Society of Paris, 1862. (See Ront- 

gen's Thermodynamics, 3, 29.) 
Preston : Theory of Heat, 36. 
Tait : Sketch of Thermodynamics, Chap. I. 
Chwolson : Lehrbuch der Physik 3, 4, 412. 



14 ENERGY [chap, i 

Conservation and Degradation of Energy 

Helmholtz : tjber die Erhaltung der Kraft. * Berlin, 1847. 

Thomson (Lord Kelvin) : Edinb. Trans. 20, 261, 289 (1851) ; Phil. Mag. (4) 

4 (1852). 
Griffiths : Thermal Measurement of Energy, Lecture 1. 
Preston : Theory of Heat, 86, 630. 
Planck : Treatise on Thermodynamics (Ogg), 40. / 

Units of Energy. The Mechanical Equivalent 

Rowland : Proc. Amer. Acad. 15, 75. 1880. 

Reynolds and Moorby : Phil. Trans. 190 A, 301. 1898. 

Schuster and Gannon: Phil. Trans. 186 A, 415. 1895. 

Barnes: Phil. Trans. 199 A, 149. 1902. Proc. Royal Soc. 82 A, 390. 

1910. 
Griffiths : Thermal Measurement of Energy. 
Chwolson : Lehrbuch der Physik 3, 414. 
Winkelmann : Handbuch der Physik 2, 537. 
Marks and Davis : Steam Tables and Diagrams, 92. 



CHAPTER II 

CHANGE OF STATE. THERMAL CAPACITIES 

12. State of a System. — A thermodyDamic system may be 
defined as a body or system of bodies capable of receiving and 
giving out heat or other forms of energy. In general, we shall 
assume such a system at rest so that it has no appreciable ki- 
netic energy due to velocity. As examples of thermodynamic 
systems, we may mention the media used in heat motors : wa- 
ter vapor, air, ammonia, etc. 

We are frequently concerned with changes of state of systems, 
for it is by such changes that a system can receive or give out 
energy. We assume ordinarily that the system is a homogeneous 
substance of uniform density and temperature throughout ; 
also that it is subjected to a uniform pressure. Such being the 
case, the state of the substance is determined by the mass, tem- 
perature, density, and external pressure. If we direct our 
attention to some fixed quantity of the substance, say a unit 
mass, we may substitute for the density its reciprocal, the vol- 
ume of the unit mass ; then the three determining quantities 
are the temperature, volume, and pressure. These physical 
quantities which serve to describe the state of a substance are 
called the coordinates of the substance. 

In all cases, it is assumed that the pressure is uniform over 
the surface of the substance in question and is normal to the 
surface at every point ; in other words, hydrostatic pressure. 
We may consider this pressure in either of two aspects : it 
may be viewed as the pressure on the substance exerted by some 
external agent, or as the pressure exerted by the substance on 
whatever bounds it. For the purpose of the engineer, the lat- 
ter view is the most convenient, and we shall always consider the 
pressure exerted hy instead of on the substance. The pressure 
is always stated as a specific pressure, that is, pressure per unit 

15 



16 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii 

of area, and is denoted by p. The unit ordinarily used is the 
pound per square foot. 

The volume of a unit weight of the substance is the specific 
volume. Ordinarily volumes will be expressed in cubic feet 
and specific volumes in cubic feet per pound. As it is frequently 
necessary to distinguish between the specific volume and the 
volume of any given weight of the substance, we shall use v to 
denote the former and V the latter. Thus, in general, v will 
denote the volume of one pound of the substance, P^the volume 
of ilf pounds ; hence 

V=Mv. 

This convention of small letters for symbols denoting quanti- 
ties per unit weight and capitals for quantities associated with 
any other weight M will be followed throughout the book. 
Thus q will denote the heat applied to one pound of gas and Q 
the heat applied to impounds, u the energy of a unit weight of 
substance, C/'the energy of il[f units, etc. 

As regards the third coordinate, temperature, we shall ac- 
cept for the present the scale of the air thermometer. Later 
the absolute or thermodynamic scale will be introduced. 
While the centigrade scale presents great advantages, the com- 
mon use of the Fahrenheit scale in engineering practice compels 
the adoption of that scale in this book. 

13. Characteristic Equation. — In general, we may assume 
the values of any two of the three coordinates p^ v^ ^, and 
then the value of the third will depend upon values of these 
two. For example, let the system be one pound of air inclosed 
in a cylinder with a movable piston. By loading the piston we 
may keep the pressure at any desired value ; then by the ad- 
dition of heat we may raise the temperature to any predeter- 
mined value. Thus we may fix p and T independently. We 
cannot, however, at the same time give the volume v any value 
we please ; the volume will be uniquely determined by the 
assumed values of p and T^ or in other words, ?; is a function 
of the independent variables p and T. In a similar manner 
we may take p and v as independent variables, in which case T 
will be the function, or we take v and T as independent and 
p as the function depending on them. 



ART. 14] 



EQUATION OF A PERFECT GAS 



17 



For any substance, therefore, the coordinates p^ v, and T are 
connected by some functional relation, as 

</> (p, V, T) = 0, (1) 

or written in the explicit form 

p=f(y,T}. (2) 

The equation giving this relation is called the characteristic 
equation of the substance. The form of the equation must be 
determined by experiment. 

For some substances more than one equation is required ; thus 
for a mixture of saturated vapor and the liquid from which it 
is formed, the pressure is a function of the temperature alone, 
while the volume depends upon the temperature and a fourth 
variable expressing the relative proportions of vapor and 
liquid. 

14. Equation of a Perfect Gas. — Experiments on the so-called 
permanent gases have given us the laws of Charles and Boyle. 
Assuming these to be fol- 
lowed strictly, we may 
readily derive the charac- 
teristic equation of a gas as 
follows. 

According to the law of 
Charles, the increase of 
pressure when the gas is 

heated at constant volume is proportional to the increase of 
temperature ; that is. 

This equation defines, in fact, the scale of the constant volume 
gas thermometer. Charles' law is shown graphically in Fig. 2. 
Point A represents the initial condition (ji?^, ^q), point B the 
final condition (jt?, ^). Then 




CB=p-p^,AC 



t-t^, and—: 



t -L 



^=k. 



According to Charles' law, therefore, the points representing 
the successive values of p and t, with v constant, lie on a straight 
line through the initial point A, and the slope of this line is the 



18 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii 

constant h. Evidently k is independent of p and t^ but it may 
depend upon v, hence we write 

Substituting this value of k in (1), we get 

In this equation t and t^ are temperatures measured from the 
Fahrenheit zero ; that is, from the origin (Fig. 2). Evidently 
the difference t— t^ is independent of the position of the as- 
sumed zero ; hence we may write 

where ^and T^ denote temperatures measured from some new 
zero, assumed at pleasure. Let us choose this new zero such 
that 2^= when p = 0. This is evidentl}^ equivalent to the 
selection of a new origin 0' (Fig. 2) at the intersection of the line 
AB with the ^-axis. If we now take the initial point A at 0', 
we have jo^ = 0, Tq = 0, and (2) takes the form 

whence pv= Tvf(y). (3) 

By hypothesis, the substance follows Boyle's law ; that is, the 
product pv is constant when the temperature T is constant. 
From (3), therefore, the factor vfQv') is a constant ; and denot- 
ing this constant by B we have 

pv = BT, (4) 

which is the characteristic equation desired. 

The name perfect gas is applied to a hypothetical ideal gas 
which strictly obeys Boyle's law, and the internal energy of 
which is all of the kinetic form, and, therefore, dependent on 
the temperature only. No actual gas precisely fulfills these 
conditions ; but at ordinary temperatures, air, nitrogen, hydro- 
gen, and oxygen so nearly meet the requirements that they 
may be considered approximately perfect. 

15. Absolute Temperature. — The zero of temperature defined 
in the preceding article is called the absolute zero, and tempera- 
tures measured from it are called absolute temperatures. The 
absolute zero may be physically interpreted as follows : By the 
kinetic theory, the pressure of a gas is due to the impact of its 



ART. 15] ABSOLUTE TEMPERATURE 19 

molecules on the containing walls. When this pressure is zero, 
we infer that molecular motion of translation has entirely ceased, 
and this is, therefore, the condition at absolute zero. 

The position of the absolute zero relative to the centigrade 
zero may be determined approximately by experiments on a 
nearly perfect gas, such as air. From Eq. (4), Art. 14, we 
have, assuming that the volume remains constant, 

whence ^ = tT' (1) 

and P^^'^1^. (2) 

Pi ^1 

Let us take T^ as the temperature of melting ice, T^ that of 
boiling water at atmospheric pressure. Regnault's experi- 
ments on the increase of pressure of air when heated at con- 
stant volume gave the relation 

^^^fyp? = 1-3665. (8) 

jt?i (at 0° C.) ^ ^ 

Since for the C. scale 

^2-1^1 = 100, 

0.3665 ». 100 
we have ^r^^ = ^^' W 

That is, using air as the thermometric substance, the abso- 
lute zero is 272.85° C. below the temperature of melting ice. 
Other approximately perfect gases, as nitrogen, hydrogen, etc., 
give slightl}^ different values for Ty The experiments of 
Joule and Thomson indicate that for an ideal perfect gas, one 
strictly obeying the law expressed by the equation pv = BT^ the 
value of T^ would be between 273.1 and 273.14. The corre- 
sponding value on the Fahrenheit scale may be taken as 491.6 ; 
that is, the absolute zero is 491. 6°f below the temperature of 
melting ice, or 459.6° below the ordinary F. zero. If then we 



20 



CHANGE OF STATE. THERMAL CAPACITIES [chap, ii 



denote ordinary temperatures by t and absolute temperatures 
by T^ we have 

T=t-\- 273.1, for the C. scale. 

T= t -h 459.6, for the F. scale. 

16. Other Characteristic Equations. — The equation pv = BT 
gives a close approximation to the changes of state of the more 
permanent gases. Other gases, as, for example, carbonic acid, 
which are in reality only slightly superheated vapors, show 
marked deviations from the behavior of the ideally perfect gas, 
and this equation does not give even a rough approximation to 
the actual facts. 

On the basis of the kinetic theory of gases, van der Waals 
has deduced a general characteristic equation applicable not 
only to the gaseous but to the liquid state as well. It has the 
following form : 

BT a .-,. 

in which B^ a, and h are constants which depend upon the 
nature of the substance. 

An empirical equation for superheated steam is 



p(v + c-)^BT-pO- + ap-)^^ 



(2) 



It will be observed that for large values of T and v, that is, 
when the gas is extremely rarified, the last term of both equa- 
tions becomes small and 
the resulting equation ap- 
proaches more nearly the 
equation of the perfect gas. 



17. Characteristic Sur- 
faces. — The characteristic 
equation 

. (^ (^, V, T) = 0, 

having three variables, may 
be represented geometri- 
cally by a surface. A state 
of the substance is defined 




ART. 18] 



THERMAL LINES 



21 



by its coordinates p^^ v^ I\, and this state is therefore repre- 
sented by a point, on the surface. If the state changes, a 
second point with coordinates jt?2, v^^ T^^ will represent the new 
state. The succession of states between the initial and final 
states will be represented by a succession of points on the 
surface. The point representing the state we will call the 
state-point. Hence, for any change of state there will be a 
corresponding movement of the state-point. 
The surface representing the equation 

pv = BT 
is shown in Fig. 3. For other characteristic equations the sur- 
faces are of a less simple form. 

18. Thermal Lines. — If we impose the restriction that during 
a change of state the temperature of the substance shall remain 
constant, the state-point will evidently move on the character- 
istic surface parallel to the jo?;-plane. Such a change of state is 
called isothermal, and the curve described by the state-point is 
an isothermal curve or, briefly, an isotherm. By taking different 
constant values for the temperature, we get a complete repre- 
sentation of the characteristic equation. For the perfect gas, 
the isotherms consist of a system of equilateral hyperbolas hav- 
ing the general equation 

p?; = const. (1) 

The restriction may be imposed that the pressure of the sub- 
stance shall remain constant during the change of state. The 
state-point will in this case move parallel to the t'^-plane, and 
the projection of the path on the ^v-plane will be a straight line 
parallel to (9F; as AB (Fig. 4). The relation between volume 
and temperature is found by 
combining the equation 

p = const. 
with the characteristic equation 
of the substance 

c\> (p, V, T) = 0. 
Thus for a perfect gas, 
pv = BT, 
and p = C. Fio. 4. 




22 



CHANGE OF STATE. THERMAL CAPACITIES [chap, ii 



Substituting this value of p in the characteristic equation, w& 
have 

B 







T. 



(2) 



If the substance changes its state at constant volume, the 
state-point moves parallel to the ^2^-plane, and the projection 
of the path on the ^v-plane is a line parallel to the j?-axis, as 
CD (Fig. 4). In the case of a perfect gas, the relation between 
p and T for a change at constant pressure is 



* 



(3) 



Lines of constant pressure are called isopiestic lines ; lines of 
constant volume, isometric lines. 

Besides the cases just given, others are of frequent occur- 
rence, and will be taken up in detail later. Thus we may have 
changes of state in which the energy of the system remains 
constant ; such changes are called isodynamic. We may also 
have changes in which the system neither receives nor gives 
out heat ; such are called adiabatic. 

19. Heat absorbed during a Change of State. — A change of 
state of a system is generally accompanied by the absorption 

of heat from external sources. 
If we denote by q heat thus 
absorbed per unit weight, we 
may by giving q proper signs 
cover all possible cases ; thus 
+ q indicates heat absorbed, ~ q 
heat rejected ; while if ^ = 0, 
we have the limiting adiabatic 
change of state. 

The heat absorbed may be 
determined from the changes in 
two of the three variables p^ v^ t that define the state of the 
system. As we have seen, any pair may be selected as suits 
our convenience. For example, let t and v be taken as the 
independent variables, and let the curve AB (Fig. 5) represent 
oh the ^v-plane a change of state. Suppose an element PR of 




Fig. 5. 



ART. 19] HEAT ABSORBED DURING A CHANGE OF STATE 23 

this curve to be replaced by the broken line PQR, then the 
segment PQ represents an increment of volume ^v with t 
constant and the segment QR an increment of temperature 
A^ with V constant. The rate of absorption of heat along PQ^ 
that is, the heat absorbed per unit increase of volume, is given 

by the derivative ( — ) , the subscript t indicating that t is held 

\dvjt 

constant during the process. If the rate of absorption be mul- 
tiplied by the change of volume v^ the product ( -^ ) Av is evi- 

\dvjt 

dently the heat absorbed during the change of state represented 

by PQ. Similarly, the rate of absorption along QR is ( — ) , 

\dtjv 

and the heat absorbed is the product [-^j A^. The heat ab- 
sorbed during the change PQR is, therefore, 

and the total heat absorbed along the broken path from A to 
B is given by the summation 



X 



dvj \dtj . 



(2) 



Ht). 



By taking the elements into which the curve is divided 
smaller and smaller, the broken path may be made to approach 
the actual path between A and B. Therefore, passing to the 
limit, we have instead of (1) 

*+(|)/^' (3) 

and for the heat absorbed during the change of state from A 

»=c:to,*Ki),]*- . « 

By choosing other pairs of variables as independent, other 
equations similar to (3) may be obtained. Thus, taking t and 
», we have . . ,^ . 



24 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii 

or taking jp and v as the independent variables, we have 

From (5) and (6) equations corresponding to (4) may be 
readily derived. 

20. Thermal Capacity. Specific Heat. — Of the partial deriv- 
atives introduced in the preceding article, two are of special 

importance, namely, ( — -) and (— ) • In general, the heat 
\dtjv \^tjp 

required to raise the temperature of a body one degree under 
given external conditions is called the thermal capacity of the 
body for these conditions. Hence, if Q denotes the heat ab- 
sorbed by a body during a rise of temperature from t^ to t^^ the 

quotient — — — gives the mean thermal capacity of the body ; 
and the quotient — - — — = , the mean thermal capacity 

of a unit weight. If the thermal capacity varies with the tem- 
perature, then the limiting value of the quotient , that 

h ~ h 

is, the derivative — ^, gives the instantaneous value of the ther- 
eto 

mal capacity. Accordingly, we recognize in the derivative 
-^ ) the thermal capacity per unit weight of the body under 
the condition that the volume remains constant; and in the 
derivative f -^ J the thermal capacity with the pressure constant. 

According to the definition of the thermal units (Art. 10), 
the thermal capacity of 1 gram of water at 17.5° C. is 1 calorie, 
and that of one pound of water at 63.5° F. is 1 B. t. u. 

The specific heat of a substance at a given temperature t is 
the ratio of the thermal capacity of the substance at this tem- 
perature to the thermal capacity of an equal mass of water at 
some chosen standard temperature. If Ave take 17.5° C. 
(63.5° F.) as the standard temperature, and denote by 7 ther- 



KatJ 



ART. 20] THERMAL CAPACITY. SPECIFIC HEAT 



25 



mal capacity per unit weight, then the specific heat c is given 
by the relation 

_ 7^(of subtance) 

717.5(0^' water)* 

But for water 7-^^ 5 = 1 cal. It follows that the specific heat at 
the temperature t is numerically equal to the thermal capacity 
of unit weight at the same temperature ; thus at 100° C. the 
thermal capacity of a gram of water is found to be 1.005 cal., 



and the specific heat is = 



7ioo 1.005 cal. 



= 1.005. On account 




Fig. 0. 



7i7.5 1 cal. 

of this numerical equality, we may consider that the derivative 

-^ represents the specific heat, as well as the thermal capacity. 

(XTj 

It is to be noted, however, that a specific heat is merely a ratio, 
an abstract number, and it is determined by a comparison of 
quantities of heat. The deter- 
mination of thermal capacity, ^ 
on the other hand, involves 
energy measurements. 

The specific heat of a sub- 
stance may be represented geo- 
metrically, as shown in Fig. 6. 
Starting from some initial state, O' 
let the rise of temperature be 
taken as abscissa and the heat added to the substance as 
ordinate. The resulting curve OM w'iVl represent the equation 

and the slope of the curve at any point, as P, will give the de- 
rivative — ^, or the specific heat at the temperature correspond- 

(XZ 

ing to P. With constant specific heat the curve OM is a 
straight line ; if the specific heat increases with the tempera- 
ture, the curve is convex to the f-axis. 

The heat applied to a substance, as will be shown presently, 
may have other effects than raising the temperature. The 
specific heat, however, is numerically the ratio of the heat 
supplied, whatever its effects upon the body, to the rise of 



26 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii 

temperature ; hence, the value of the specific heat will depend 
upon the conditions under which the heat is absorbed. If the 
substance is in the solid or in the liquid form, the specific heats are 
practically equal. For substances in the gaseous form, however, 
the specific heat may have any value from — oc to + oc, depending 
upon the external conditions under which the heat is supplied. 

21. Latent Heat. — If the heat added to a substance and the 
temperature be plotted as in Fig. 6, it may happen that at cer- 
tain temperatures the curve has discontinuities. For example, 

let heat be applied to ice 
at 0° F. The curve is 
practically a straight line 
until the temperature 32° 
is reached, but at this 
point considerable heat is 
added without any change 
in temperature. During 
this addition of heat, rep- 
resented by the vertical 
Pj(. rj^ segment AB (Fig. 7), the 

state of aggregation 
changes from solid to liquid. As the water receives heat its 
temperature rises, as indicated by BC, until tlie temperature 
212° F. is reached (assuming atmospheric pressure), where the 
temperature again remains constant during the addition of a 
considerable quantity of heat, and the state of aggregation again 
changes, this time from the liquid to the gaseous. The heat 
that is thus added to (or abstracted from) a substance during 
a change of state of aggregation is called latent heat. As 
pointed out in Art. 4, substantially all of the latent heat is 
stored in the system in the form of potential energy. 

The specific heat -^ becomes infinite during the changes 

indicated by AB and CD, since t = constant. The volume of 
the substance changes, however, and the rate at which heat is 

added with respect to the volume, that is, the derivative f ^ 




ART. 22] RELATIONS BETWEEN THERMAL CAPACITIES 27 

is a thermal capacity called the latent heat of expansion and 
denoted by l„. If the pressure also changes, we have in the 

derivative (|)^ the heat added per unit ehange of pressure. 

This thermal capacity is called the latent heat of pressure varia- 
tion, and is denoted by l^. 

22. Relations between Thermal Capacities. — Introducing the 
symbols c^, c^, l^, and l^ in equations (3) and (5) of Art. 19, we 
have 

dq = l^dv + CydT^ ' (1) 

dq = Ipdp + CpdT. (2) 

By means of the characteristic equation of the substance, 
namely, 

v=fCT,p-), (3) 

various relations between the thermal capacities may be de- 
rived. Some of the most useful are the following. 
From (3) we obtain by differentiation, 

dv = ^dT^^-^dp, (4) 

dT dp ^ ^ ^ 

which substituted in (1) gives 

dq=l,^^dp+{c, + l,^dT. (5) 

Comparing (2) and (4), we have 

l^^K^, (6) 

Cp-o^^lvj^' (T) 

In the same way, substituting 

dp=%dT+fdv 

^ dT dv 

in (2), and comparing the resulting equation with (1), we 
obtain 

K=i/i, (8) 



28 CHANGE OF STATE. THERMAL CAPACITIES [chap, n 

The relations thus obtained enable us to calculate the remain- 
ing thermal capacities when any one is given by direct experi- 
ment, provided the characteristic equation of the substance is 

dv dp 
known, so that the derivatives — ^, --^, etc., can be determined. 

o 1 o 1 

For a perfect gas, as an example, c^ is known from experiment 
and the ratio — has also been determined. From the equation 
of the gas pv = BT^ we have the partial derivatives 
dv _B dp_B^ 

hence from (7) and (9) 

B p 

c^-€^ = l or l^ = ^(Cp- tO , (10) 

and h = -^C^p- ^v)' (11) 

23. Interpretation of Differential Expressions. — In thermo- 
dynamics we frequently meet with expressions of the form 

Mdx 4- JSfd^ 

composed of two terms, of which each is the differential of a 
variable multiplied by a coefficient. The two coefficients may 
be constants or functions of the two variables involved. The 
proper interpretation of differentials of this form is likely to 
present difficulties to the student ; we shall, therefore, devote 
this article to a discussion of such expressions, their properties, 
and their physical interpretations. 

Let us consider first how such differential expressions may 
arise. Suppose we have given the characteristic equation of a 
substance in the form 

p=/C<',0; (1) 

by differentiation according to the well-known methods of cal- 
culus, we obtain the relation 

dp = t^dv + %dt, (2) 



ART. 23] DIFFERENTIAL EXPRESSIONS 29 

which may be written in the form, 

dp = Mdv + Ndt, (3) 

where M= ^, and ]V=-^' 
dv dt 

In Art. 19 we derived an equation of similar form, namely, 
dq = ^dv+^^dt, (4) 

which may likewise be written in the form 

dq = M'dv + JST'dt. (5) 

The second members of (3) and (5) are differential expressions 
of the form Mdx + JVd^^ which we have under consideration. 
Eq. (3) was produced from a known functional relation be- 
tween j9, V, and ^, while Eq. (5) was derived directly from 
physical considerations by assuming increments Av and A^ of 
the independent variables and deducing from them the quantity 
of heat Aq that must necessarily be absorbed. No relation 
between g, v, and t was given or assumed ; in fact, it is known 
that no such relation exists ; that is, q cannot be expressed as a 
function of the variables v and t. 

Let us see what is implied by the existence or non-existence 
of a functional relation between q, v, and t. Referring to 
Fig. 5, let A and B denote the initial and final states of the 
system. Since |? is a function of v and t [p —f(y^ 0]? the 
pressures at A and B are determined by the values of T and v 

at those points ; thus for a perfect gas, p-^ = 1 and p^= ^. 

Hence, the change of pressure p^ — P\ ^^ passing from Ato B 
is fixed by the points A and B alone and is independent of the 
path between them. Similarly, if there is a functional rela- 
tion between q, v, and ^, that is, ii q = c^) (v, ^), we shall have at 
A, ^^ = c^(v^, ^j), at B, g2 = <^(^2' ^2)- Therefore, the heat 
absorbed in passing from Aio B will be 

92-9i = 't> (^2' ^2) - ^ (^1^ ^1)' (^) 

and this will be determined by the points A and B alone. On 
the other hand, if the heat absorbed by the system depends 
upon the path between A and B, there can be no relation 



30 



CHANGE OF STATE. THERMAL CAPACITIES [chap, ii 



q = ^('t;, f). As a matter of fact, the lieat absorbed is different 
for different paths between the same initial and final states ; 
hence it. is not possible to express q in terms of v and t. 

The conclusions just given may be stated in general terms as 
follows. Given an expression of the form 

du = Mdx-^ Ndy, , (7) 

where the coefficients M and N are functions of x and ?/, there 
may or may not exist a functional relation between u and the 
variables x and y. If ?^ is a function of x and y^ say u = FQc^ ^), 
then the change in u depends only on the initial and final 
values of x and y and is independent of the path. This change 
is found from (7) by integration ; thus 



r^ du = f "^' ^=^ (Mdx + JVdy^ . 



(8) 



In this integration no relation between x and y is required, for 
since Mdx + JVdy arises from differentiating the function 
(j) (x^ y), the integral must be (/> (rr, ?/). In this case Mdx -\-Ndy 
is said to be an exact differential. 

As an example, consider the equation 

du — ydx + xdy. 

Since ydx + xdy is produced by the differentiation of the prod- 
uct xy^ we have the relation 

u — xy-\-C^ 
whence u^— u^z= x^y^ — x^y^ 

The change of u is represented by the shaded area (Fig. 8), 

and is evidently not dependent 
upon the path between the points 
ioc^rV^) (^1, «/i) and (x^, y^. 

If, however, no functional rela- 
tion exists between u and the 
variables x and ?/, then Mdx + 
Ndy is said to be an inexact 
differential. In this case a value 
of u cannot be found until a 
relation between x and y is as- 




ART. 23] DIFFERENTIAL EXPRESSIONS 31 

sumed ; in other words, the v^lue of u depends upon the path 
between the initial and final points. For example, let 

du = ydx — 2 xdy 
and let the initial and final points be respectively (0, 1) and 
(2, 2). No function of x and y can be found Avhich upon 
differentiation will produce this differential. If we choose as 
the path between the end points the straight line y = \x-\-l^ 
we have (since dy = | dx^^ 

u=\ \^(^x-{-\)dx — xdx^=l. 

If we take as the path the parabola y = \x^ -\-l^ we have 

u = p[(i x^ + l')dx - xMx'] = 0. 

The dependence of the value of u upon the path assumed is 
evident. 

The test for an exact differential is simple. If the differential 
du = Mdx + Ndy is exact, then u must be a function of x and y^ 
say /(a;, ^). By differentiation, we have 

^ du J du J 

au= — ax-\ dy. 

dx dy 

Hence 31 and W must be, respectively, the partial derivatives 

du du 

—- and ^— • By a well-known theorem of calculus, we have 

dx dy 

d_fdu\^±fdu\ 

dy\dxj dx\dyj^ 

that IS, -_ = -—. (9) 

dy dx 

If relation (9) is satisfied, the differential is exact ; otherwise, 
it is inexact. 

As an example, we have from the differential ydx — 2 xdy^ 

— — = 1, — = — 2 ; therefore, the differential is inexact, as was 
dy dx 

shown in the preceding discussion. 

In thermodynamics we meet with certain functions that de- 
pend only upon the coordinates p, f , T of the substance under 
consideration. From purely physical considerations the energy 
u of the substance is known to be a function of the state only. 



32 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii 

(See Art. 26.) Hence if u is expressed in terms of two of 
these coordinates as independent variables, thus, 

du = Mdv + JYdT, 

we know at once that du is exact and we can write 

Furthermore, from the test for an exact differential we must 
have the relation 

dT dv' 

By making use of this test when the differential is known to 
be exact, many useful relations are deduced. 

We have also magnitudes that depend upon the coordinates 
and also upon the method of variation ; that is, upon the path. 
The heat q absorbed by a system in changing state is one of 
these. If again we choose v and T as the independent variables, 
we may write 

dq = M'dv 4- JST'dT', 

but since dq is not exact, we cannot write 

^^dq = q^- q^. 

EXERCISES 

1. Regnault's experiments on the heating of certain liquids are ex- 
pressed by the following equations : .^ 

Ether q = 0.529 t + 0.000296 t'^, - 20° to + 30° C. 

Chloroform q = 0.232 t + 0.0000507 t\ - 30° to + 60° C. 

Carbon disulphide q = 0.235 t + 0.0000815 t'^, - 30° to + 40° C. 
Alcohol q = 0.5476 t + 0.001122 i^+ 0.0000022 t% - 23° to + 66° C. 

From these equations derive expressions for the specific heat, and for 
each liquid find the specific heat at 20° C. 

2. From the data of Ex. 1, find the mean heat capacity of ether between 
0° and 30° C. Also the mean heat capacity of alcohol between 0° and 50° C. 

3. If the thermal capacity of a substance at temperature t is given by 
the relation 

y = a + ht + ct% 

show that the mean thermal capacity between 0° and t is given by the relation 



ART. 23] EXERCISES 33 

4. In the investigation of the properties of gases, it is convenient to 
draw the isothermals {T := const.) on a plane having the pressure p as the 
axis of abscissas and the product pv as the axis of ordinates. Show that 
the isothermals of a perfect gas are straight lines parallel to the jo-axis. 

5. Show on the pv-p plane the general form of an isothermal of super- 
heated steam, the characteristic equation being 

p{v + c)^BT-p{l-^-ap):^^. 

As an approximate equation for superheated steam, the form 

p{v + c)^BT, 

has been suggested by Tnmlirtz. Show the form of the isothermal when 
this equation is used. 

6. Derive relations between c^, c„, Ip, and U, similar to those given by 
Eq. (10) and (11) of Art. 22, using van der Waal's equation 

V — v^ 
as the characteristic equation of the gas. 

7. For a perfect gas, as will be shown subsequently, the thermal capacitiy 
ly is Ap{A — i). Show that c^ — c„ = AB ; also that Ip = — Av. 

8. Test the following differentials for exactness : 

(a) vdp + npdv. 

(b) v^dp + npv'^-hlv. 

9. Find the function u =f(p, T) which produces the differential (c) 
of Ex. 8. 

dT r 

10. The differential [c'(l — x)-{- c"x'\ —--\- — dx, which appears in the 

discussion of vapors, is known to be exact, c' and c" may be taken as con- 
stants, while r is a function of T. Apply the test for exactness and thereby 

deduce the relation c" — c' = — ,• 

dT T 

11. For perfect gases, dq = c^dT + Apdo. (See Ex. 7, and Art. 22.) 
Making use of the characteristic equation pv = BT, show that while dq is 

not an exact differential, -? is an exact differential. 
T 

REFERENCES 

Thermal Capacity. Specific Heat 

Preston : Theory of Heat, 211. 

Griffiths : Thermal Measurement of Energy, 95. 



34 CHANGE OF STATE. THERMAL CAPACITIES [chap, ii 

Weyrauch : Grundriss der Warme-Theorie 1, 60. 
Chwolson : Lehrbuch der Physik 3, 172. 

Exact and Inexact Differentials in Thermodynamics 

Chwolson : Lehrbuch der Physik 3, 434. 

Clausius : Mechanical Theory of Heat, Introduction. 

Preston : Theory of Heat, 597. 

Weyrauch : Grundriss der Warme-Theorie 1, 28. 

Townsend and Goodenough : Essentials of Calculus, 245. 



CHAPTER III 

TEE FIRST LAW OF THERMODYNAMICS 

24. Statement of the First Law. — The first law of Thermo- 
dynamics relates to the conversion of heat into work, and merely 
applies the principle of conservation of energy to that process. 
It may be formally stated as follows : When work is expended 
in producing heat^ the quantity of heat generated is proportional to 
the work done, and conversely, when heat is employed to do work, a 
quantity of heat precisely equivalent to the work done disappears. 

If we denote by Q the heat converted into work and by IT the 
work thus obtained, we have, therefore, as symbolic statements 
of the first law, 

W^ JQ, ovQ = AW. 

25. Effects of Heat. — When a thermodynamic system, as a 
given weight of gas or a mixture of saturated vapor and liquid, 
undergoes a change of state, it in general receives or gives out 
energy either in the form of heat or in the form of mechanical 
work. These energy changes must, of course, conform to the 
conservation law. Suppose in the first place that the system is 
subjected to a uniform external pressure and that during the 
change of state the volume is decreased. Mechanical work is 
thereby done upon the system, or in other words, the system 
receives energy in the form of work. At the same time heat 
may be absorbed by the sj^stem from some external source. 
Denoting by AIT the work received and hj AQ the heat 
absorbed, the increment AZ7 of the intrinsic energy of the 
system is given by the relation 

AU = JAQ + AW. (1) 

Ordinarily we take the work done by the system in expanding 
as positive ; hence the work done on the system during com- 
pression is negative and (1) takes the form 

AU = JAQ - AW; (2) 

35 



36 THE FIRST LAW OF THERMODYNAMICS [chap, iir 

that is, the increase of energy of the system is equal to the 
energy received in the form of heat less the energy given to 
the surrounding systems in the form of work. We may also 
write (2) in the form 

JAQ =AU -^ AW, (3) 

and interpret the relation as follows. The heat absorbed by a 
substance is expended in two ways : (1) in increasing the 
intrinsic energy of the substance ; (2) in the performance of 
external work. 

Equation (3) is the energy equation in its most general form. 
Any one of the three terms may be positive or negative. We 
consider AQ positive when the system absorbs heat, negative 
when it gives out heat ; as before stated, A TF is positive when 
work is done % the system, negative when work is done on the 
system; ACT is positive when the internal energy is increased, 
negative when the energy is decreased during the change of 
state. 

26. The Intrinsic Energy. — The increase A Uoi the intrinsic 
energy is, in general, separable into two parts : (1) The in- 
crease of kinetic energy indicated by a rise of temperature of 
the system. As we have seen, this is due to an increase in the 
velocity of the molecules of the system. (2) The increase of 
potential energy arising from the increase of volume of the 
system. To separate the molecules against their mutual attrac- 
tions, or to break up the molecular structure, as is done in 
changing the state of aggregation, requires work, and this 
work is stored in the system as potential energy. 

The energy U contained in a body depends upon the state 
of the body only, and the change of energy due to a change 
of state depends upon the initial and final states only. In 
Fig. 9, let A represent the initial, and B the final state. The 
point B indicates a definite state of the body as regards pres- 
sure, volume, and temperature. Now the temperature indi- 
cated by B fixes the kinetic energy and the volume at B 
determines the potential energy. Hence the final total energy 
depends upon the coordinates of B and in no way upon the 
intermediate process represented by the path leading from A 



ART. 27] EXTERNAL WORK 37 

to B. Whether we pass by the 
path m or the path n^ we have the 
same volume and temperature at B 
and therefore the same total energy. 
Since U is thus a function of the 
coordinates only, it follows that d U 
is always an exact differential. 

Choosing T and v as the inde- ^ p^^, 9 

pendent variables of the system, 

we may express Z7 as a function of these variables. We have, 
therefore, U^f(T, v\ 

whence dU= —-dT -\- dv, 

dT dv 

r) TT 

The term —^dT is the increment of energy due to the in- 

crease of temperature (il^. The factor-— is the rate at which 

the energy changes with the temperature when the volume 

5 TJ 
remains constant. Hence ——dT is the change of energy due 

merely to the rise of temperature, that is, it is the change 
of kinetic energy. The term ■ dv is the change of energy 

due merely to the change of volume with the temperature 
constant ; it is, therefore, the work done against molecular 
attractions, the work that is stored as potential energy. For 
a substance in which there are no internal forces between 
the molecules, the energy is independent of the volume, that 

is, = 0, and therefore the term dv is zero. 

bv dv 

27. The External Work. — In nearly all cases dealt with in 
applied thermodynamics, the external work A TF is the work 
done by the system in expanding against a uniform normal 
pressure. A general expression for the external work may 
be deduced as follows. Let AJ^ denote an elementary area 
on the surface inclosing the system and suppose that during 
the expansion of the system this area moves in the direction 
of the normal to it through a distance s. If then p is the 



38 



THE FIRST LAW OF THERMODYNAMICS [chap, hi 



normal pressure per unit area, the work done against this 
pressure is for this one element 

p^Fs. (1) 

When all the elements of the surface are taken, the expres- 
sion for the work is 

^W=p^s^F. , (2) 

But evidently if s be taken sufficiently small, 2s A^ is the 
increase of volume AF"; hence we may write 

ATr=jt?AF; (3) 



from which we have 



w==^p^v=Qpdv 



(4) 



for a change of volume from V^to V^. 

The external work for a given change of state is represented 
graphically by the area between the projection of the path 
of the state-point on the p F'-plane and the F'-axis. Thus in 
Fig. 10, let the variation of pressure and volume be represented 
by the curve AB ; this is the projection on the p F-plane of the 
actual path of the state-point on the characteristic surface. 
The area A -^ABB^ under AB is clearly given by the integral 

hence, it represents the work done by the system in passing from 
the initial to the final state according to the given law. 

The general energy equa- 
tion (3), Art. 25, may now be 
written in the form 

JAQ=AU^pAV, (5) 
or using the differential nota- 
tion, in the form 

JdQ = dU-{-pdV. (6) 
For a unit weight of the sub- 
stance, we have 

Jdq = du-{- pdv. (6 a) 

28. Integration of the Energy Equation. — The heat imparted 
to a system during a change of state is found by integrating 




ART. 29] ENERGY EQUATION 39 

the equation just derived. Denoting the initial and final states 
by the subscripts 1 and 2, respectively, we have 

whence JQ= U^~ U^-\- ^ p dV (1) 

It should be noted carefully that since the energy U depends 
only upon the state of the system and not upon the process of 
passing from the initial to the final state, the change of energy 
may be written at once as the difference C/g — U^ The external 
work 

W= ^^dV 

is evidently dependent upon the path of the state-point between 
the initial and final states. See Fig. 10. Hence the sum of 
the change of energy and external work, that is, the heat added 
to the system, must also depend upon the path. It follows 
that dQ is not an exact differential, and we cannot write 

^^dQ=Q^-Q,. 

In other words, we cannot properly speak of the heat in a 
a body in the state 1 or the state 2 ; we can speak only of the 
heat imparted to the body during the change of state with the 
reservation, stated or implied, that the quantity thus imparted 
depends upon the way in which the state is changed. For con- 
venience we shall denote by Q-^^^ ^^^^ heat imparted to the sys- 
tem in passing from state 1 to state 2 ; and likewise by W12 the 
corresponding external work done by the system. 

29. Energy Equation applied to a Cycle Process. — Let a sys- 
tem starting from an initial state pass through a series of pro- 
cesses and finally return to the initial state. The path of the 
state-point on the characteristic surface is a closed curve in 
space and the projection of the path on the p F^plane is a closed 
plane curve. See Fig. 11. Let A represent the initial state; 
then in passing from A to B the external work done by the 
system is 

\ ^ p c^F" (along path m), 



40 



THE FIRST LAW OF THERMODYNAMICS [chap, hi 




^1 



Fig. 11. 



which is represented by area A^AmBB^, while in passing from 
B back to A along path n the external work is 

J^ pdV= — )y P ^F (along path 9^), 

and this is represented by area B^BnAA^ Hence the net 
external work done by the system is represented by the area 

inclosed by the curve of the 

cycle. 

Since the energy U of the 

system depends upon the state 

only, the change of energy for 

the cycle is 

Y and the energy equation re- 
duces to 

That is, for a closed cycle of processes^ the heat imparted to the 
system is the equivalent of the external work^ and both are repre- 
sented graphically by the area of the cycle on the pF^plane. 

30. Adiabatic Processes. — When a system in changing its 
state has no thermal communication with other bodies and 
therefore neither absorbs nor gives out heat, the change of 
state is said to be adiabatic. In general, adiabatic changes are 
possible only when the system is inclosed in a non-conducting 
envelope. Rapid changes of state are approximately adiabatic, 
since time is required for conduction or radiation of heat ; thus 
the alternate expansion and contraction of air during the pas- 
sage of sound waves is nearly adiabatic : the flow of a gas or 
vapor through an orifice is practically an adiabatic process. 

For an adiabatic change, the term JQ of the energy equation 
reduces to zero, and we have, consequently. 







or 



w. 



12 



U,~U,. (1) 

During an adiabatic change, therefore, the external work done 
by the system is gained at the expense of the intrinsic energy 
of the system. 



ART. 30] 



ADIABATIC PROCESSES 



41 



The projection on the joT^-plane of the path of the state-point 
during an adiabatic change gives the adiabatic curve. See Fig. 
12. The area A^ABB^ represents the work W12 of the system 
and from (1) it represents also the decrease of the intrinsic 
energy in passing from state 1 
represented by A to state 2 
represented by B. Making 
use of this principle, we can 
arrive at a graphical represen- 
tation of the intrinsic energy 
of a system. Suppose the 
adiabatic expansion to be con- 
tinued indefinitely ; the adia- 
batic curve AB will then 
approach the F^axis as an 

asymptote, and the work of the expanding system will be 
represented by the area A^A oo between the ordinate A^A^ the 
axis OFJ and the curve extended indefinitely. The area A^A 00 
represents also the change of energy resulting from the expan- 
sion. Hence if we assume that the final energy is zero, we have 

Z7j — = area A^A 00, 




Ax 



B, 



Fig. 12. 




or 



JJ^ = area A^A 00 



= 1. 



P 



dV. 



A, 



Fig. 13. 



It is instructive to compare 
the adiabatic curve with the 
isothermal. When the two 
curves are projected on the 
pF-plane, the adiabatic is the 
^ steeper. See Fig. 13. This 
follows from the fact that dur- 
ing adiabatic expansion the 
energy decreases and as a result the temperature falls ; hence 
for the same final volume F^, the temperature, and therefore 
the pressure, is lower for the adiabatic expansion than for the 
isothermal expansion. 

If the two curves are produced indefinitely, each will have the 
axis OF' as an asymptote. The area under the adiabatic, which 
represents the initial intrinsic energy of the system, is finite. 



42 



THE FIRST LAW OF THERMODYNAMICS [chap, m 



On the other hand, the area under the indefinitely extended 
isothermal is infinite. 

31. Isodynamic Changes. — If the intrinsic energy of the 
system remains unchanged during a change of state, the change 
is called isodynamic or isoenergic. In this case the energy 
equation reduces to the form 



JQ 



12 



tf;o = 



12 






dV. 



For perfect gases, the isodynamic curve is also the isothermal, 
but for other substances this is not the case. 

32. Graphical Representa- 
tions. — The three magnitudes 




^§12. ^2 



C^j, and 



TF^g enter- 



ing into the energy equation 
can be represented graphically 
by areas on the joF^plane. 
Suppose the change of state 
to be represented by the curve 
m between the initial point A 
and final point B (Fig. 14). 
Let adiabatic lines be drawn 
through A and B and 



ex- 



tended indefinitely; then from preceding considerations we have 

TFi2 = area A^ABB^, 
U^ = area A^A oo , 
U^ = area B^B oo . 
Hence, JQ^^ = U^- U^j^ W-^^ 

= area A^ABB^ + area B^B oo — area A^A oo 
= area AB oo . 

That is, the heat imparted is represented on the p V-plane hy the 
area included between the path and two indefinitely extended adia- 
batics drawn through the initial and final points, respectively. 

Through the initial point A let an isodynamic be drawn, 
cutting BB^ in the point C, and through let the indefinitely 
extended adiabatic C ao he drawn. Then the energy C/g of the 
system in state O is equal to ZJ^, and, therefore, 

^3 = area B^B oo — area B^O co 
area oo CB oo . 



u^-ir,= u. 



ART. 32] 



GRAPHICAL REPRESENTATIONS 



43 



It should be noted that the 
area representing U^ — U-^ is 
not influenced by the path m. 
A second graphical repre- 
sentation is shown in Fig. 15. 
Through the initial point A 
an isodynamic line is drawn, 
and through the final point B 
an adiabatic is drawn, the two 
lines intersecting at point C. 
We have then, denoting the 
energy in the state C by ZZg, 

TJ^ _ jj^= jj^_ jj^^ area B^BCG^, 
W[^ = area A^ABB^, 
JQ^^ = W^^ -\-U^-U^ = area A^ABOC^, 

As before, the change of energy is independent of the path w, 
while both the external work and the heat imparted depend 
upon the form of m. 




EXERCISES 

1. Show that the energy equation may be written in the form 



'"'-[ff)/^-V^i)M 



dv. 



and that consequently the derivative f — j must be equal to Jc„. 

2. If the energy of a substance is independent of the volume, show that 
the energy equation reduces to the form 

Jdq = Jcyd T + pdv. 

3. Using the method of graphical representation, show by areas Q12, 
U2 — Ui, and W12 (a) for a change at constant pressure, (&) for a change at 
constant volume. 

4. Show graphically that y^, the heat capacity of a gas at constant 
pressure, is greater than y„, the heat capacity at constant volume. 

5. Derive an expression for the external work done when a gas (or vapor) 
is heated at constant pressure. 

6. Derive an expression for the external work when a gas expands from 
a volume V^ to a volume V2 according to the lawpF" = const. 



44 THE FIRST LAW OF THERMODYNAMICS [chap, in 

7. Apply the general energy equation to the process of changing ice at 
32° F. to water. What is the effect of greatly increasing the pressure on 
the ice during the process ? 

REFERENCES 

Preston : Theory of Heat, 596. 

Zeuner : Technical Thermodynamics (Klein) 1, 28. 

Planck: Treatise on Thermodynamics (Ogg), 38. 



CHAPTER IV 

THE SECOND LAW OF THERMODYNAMICS 

33. Introductory Statement. — While the first law of thermo- 
dynamics gives a relation that must be satisfied during any 
change of state of a system, and of itself leads to many useful 
results, it is not sufficient to set at rest all questions that may 
arise in connection with energy transformations. It gives no 
indications of the direction of a physical process ; it imposes no 
conditions upon the transformations of energy from one form to 
another except that there shall be no loss, and thus gives no in- 
dication of the possibilities of complete transformation of dif- 
ferent forms; it furnishes no clue to the availability of energy 
for transformation under given circumstances. To settle these 
questions a second principle is required. This principle, called 
the second law of thermodynamics, has been stated in many ways. 
In effect, however, it is the principle of degradation of energy, 
just as the first law is the principle of the conservation of 
energy. 

There are conceivable processes which, while satisfying the 
requirements of the first law, are declared to be impossible be- 
cause of the restrictions of the second law. As a single ex- 
ample, it is conceivable that an engine might be devised that 
would deliver work without the expenditure of fuel, merely by 
using the heat stored in the atmosphere; in fact, such a device 
has been several times proposed. The first law would not be 
violated by such a process, for there would be transformation, 
not creation of energy; in other words, such an engine would 
not be a perpetual motion of the first class. Experience shows, 
however, that a process of this character, while not violating 
the conservation law, is nevertheless impossible. The statement 
of its impossibility is, in fact, one form of statement of the 
second law. 

45 



46 THE SECOND LAW OF THERMODYNAMICS [chap, iv 

34. Availability of Energy. In Art. 8 was noted the distinc- 
tion between various forms of energy with respect to the pos- 
sibility of complete conversion. We shall now consider the^ 
point somewhat in detail. 

Mechanical and electrical energy stand on the same footing 
as regards possibility of conversion; either can be completely 
transformed into the other in theory, and nearly so in practice. 
Either mechanical or electrical energy can be completely trans- 
formed into heat. On the other hand, experience shows that 
heat energy is not capable of complete conversion into mechan- 
ical work, and to get even a part of heat energy transformed 
into mechanical energy, certain conditions must be satisfied. 
As a first condition, there must be two bodies of different tem- 
perature ; it is impossible to derive work from the heat of a body 
unless there is available a second body of lower temperature. 
Suppose we have then a source S at temperature T^ and a re- 
frigerator H at lower temperature T^; how is it possible to 
derive mechanical work from a quantity of heat energy Q-^ stored 
in aS'? If the bodies /S' and M are placed in contact, the heat 
Q-^ will simply flow from S to H and no work will be obtained. 
Hence, as a second condition, the systems iS'and R must be kept 
apart and a third system M must be used to convey energy. 
This third system is the working fluid or medium. In the steam 
plant, for example, the boiler furnace is the source a9, the con- 
denser is the refrigerator i^ at a lower temperature, and the 
steam is the medium or working fluid M. The medium M 
is placed in contact with S and receives from it heat Q^; it then 
by an appropriate change of state (expansion) gives up energy in 
the form of work, and delivers to i2 a quantity of heat Q^', 
smaller than $-^, the difference Q^ — Q^ being the heat trans- 
formed into work. The details of this process will be given in 
following articles, where it will be shown that in no other way 
can a larger fraction of the heat be transformed into work. 

The part of the heat Q-^ that can be thus transformed into work, 
that is, Q^ — §21 is t^® available part of Q-^; and the part §2 ^^^^ 
must be rejected to the refrigerator i?, and which is of no further 

use, is the unavailable part of Q-^^ or the waste. The ratio -^^ — ^ 



ART. 35] 



REVERSIBILITY 



47 



is called the availability of Q^ for transformation into mechani- 
cal work. In general, the term availability signifies the fraction 
of the energy of a given system in a given state that can be 
transformed into mechanical work. 

In Art. 8 attention was called to the apparent tendency of 
energy to degenerate into less available forms. We have now 
to investigate this point somewhat closely in connection with 
reversible and irreversible changes of state. 

35. Reversibility. — The processes described in thermo- 
dynamics are either reversible or irreversible. A process is 
said to be reversible when the following conditions are fulfilled : 

1. When the direction of the process is reversed, the system 
taking part in the process can assume in inverse order the 
states traversed in the direct process. 

2. The external actions are the same for the direct and re- 
versed processes or differ by an infinitesimal amount only. 

3. Not only the system undergoing the change but all con- 
nected systems can be restored to initial conditions. 

A process which fails 
to meet these require- 
ments in any particular 
is an irreversible pro- 
cess. The following 
examples illustrate the 
above definitions. 

(1) Suppose a con- 
fined gas to act on a 
piston, as in the steam 
or gas engine. See 
Fig. 16. If A is the 
piston area, the pres- 
sure acting on the face 
of the piston is pA, 
and for equilibrium 

this pressure must be equal to the force F. . If now Ave assume 
the force pA slightly greater than #, the piston will move 
slowly to the right and the confined gas will assume a succes- 




F 



48 THE SECOND LAYf OF THERMODYNAMICS [chap, iv 

sion of states indicated by the carve AB. If at the state B 
the motion is arrested and F is made intinitesimally greater 
than pA for all positions of the piston, the series of states from 
B to A will be retraced and the system (the confined gas in 
this case) will be brought back to its original state without 
leaving changes in outside bodies. The reversed process is 
accomplished by an infinitely small modification of tile external 
force JF. The process is therefore reversible. 

(2) Let the force F be removed entirely. Then the piston 
will move suddenly and the confined gas will be thrown into 
commotion. When the gas finally attains a state of thermal 
equilibrium with the volume Fg, that state will be represented 
by some point as B^. No path can be drawn between A and B 
because during the passage from A to B' the gas is not in 
thermal equilibrium, and its state at any instant cannot, there- 
fore, be determined. Evidently, therefore, the gas cannot be 
returned to state A by reversing in all particulars the direct 
change from A to B'. It can be returned to state A, however, 
in the following manner : A force F^ slightly greater than pA, 
is applied to the piston and the gas is thus compressed slowly, 
the successive states being indicated by the curve B'A', say. 
Then the gas in the state A' is cooled at the constant volume 
V\ until the original state A is attained. The restoration of 
the gas to its initial state has, however, left changes in other 
bodies or systems. Thus the work of compression from B' to 
A' must be furnished from one external body, and the heat 
given up by the cooling from A' to A must be absorbed by 
another external body. The free expansion of the gas is, 
therefore, an irreversible process. 

It is easy to see that the flow of a fluid through an orifice 
from a region of high pressure to a region of low pressure is 
essentially equivalent to the irreversible expansion just de- 
scribed. Such cases are of frequent occurrence in technical 
applications of thermodynamics. The flow of liquid ammonia 
through the expansion valve of the refrigerating machine may 
be cited as an example. 

(3) The direct conversion of work into heat is an irreversi- 
ble process. For example, consider the heating of a journal 



ART. 36] GENERAL STATEMENT OF THE SECOND LAW 49 

and bearing due to the conversion into heat of the work of 
overcoming friction. A complete reversal of this process would 
involve turning the shaft in the opposite direction by cooling 
the bearing. 

(4) The conduction of heat from one body to another is an 
irreversible process. There must be a temperature difference 
to produce the flow of heat, and heat of itself will not flow in 
the reverse direction ; that is, from the colder to the hotter 
body. If, however, we take the temperature difference A2^ in- 
definitely small and let the transfer take place very slowly, the 
process can be reversed by changing the sign oiAT. Hence 
we can conceive of reversible flow as the ideal limiting condi- 
tion of the actual irreversible flow. 

Strictly speaking, there are no reversible changes in nature. 
We must consider reversibility as an ideal limiting condition 
that may be approached but not actually attained when the 
processes are conducted very slowly. 

36. General Statement of the Second Law. — According to 
the first law, the total quantit}^ of energy in a system of bodies 
cannot be increased or decreased by any change, reversible or 
irreversible, that may occur within the system. It is not, how- 
ever, the total energy, but the available energy of the system 
that is of importance ; and experience shows that a change 
within the system usually results in a change in the availability 
of the energy of the system. 

It may be considered as almost self-evident that no change 
of a system which will take place of itself can increase the 
available energy of the system. On the other hand, experience 
teaches that all actual changes involve loss of availability. Con- 
sider, for example, the flow of heat from a body of temperature 
I\, to another at temperature T^. For the flow to occur of it- 
self we must have T^ > T^^ and as a result of the process there 
is a loss of availability. To produce an increase of availability 
would require T^ to be greater than T^ ; in that case, however, 
the process would not be possible. In the limiting reversible 
case, 2^2 = ^i» ^he availability remains unchanged. A consid- 
eration of other physical processes within the range of experi- 



50 THE SECOND LAW OF THERMODYNAMICS [chap, iv 

ence leads to similar results. We may, -therefore, lay down 
the following general laws, which, like the law of conservation 
of energy, are based entirel}' on experience: 

I. No change in a system of bodies that can take place of itself 
can increase the available energy of the system. 

II. An irreversible change causes a loss of availability. 

III. A reversible change does not affect the availability. 
These statements may be regarded as fundamental natural 

laws underlying all physical and chemical changes- The second 
and third together constitute the law of degradation of energy. 
The first may be taken as a general statement of the second law 
of thermodynamics. 

By considering special processes the general statement of the 
second law here given may be thrown into special forms. Thus 
if heat could of itself pass from a body of lower to a body of 
higher temperature, the result of the process would be an in- 
crease of available energy, a result that is impossible according 
to oar first statement. We have, therefore, Clausius' form of 
the second law, viz : 

It is impossible for a self-acting machine unaided by any exter- 
nal agency to convey heat from one body to another at higher 
temperature. 

Again, if we consider the increase of available energy that 
would result from deriving work directly from the heat of the 
atmosphere, we are led to Kelvin's statement, namely : 

It is impossible by means of inanimate material agency to derive 
mechanical effect from any portion of matter by cooling it below the 
temperature of surrounding objects. 

In order to estimate the available energy of a system in a 
given state, or the loss of available energy when the system 
undergoes an irreversible change, it is necessary to know the 
most efficient means of transforming heat into mechanical work 
under given conditions. This knowledge is furnished by a 
study of the ideal processes first described by Carnot in 1824. 

37. Carnot's Cycle. — Suppose that the conditions stated in 
Art. 34 are furnished ; that is, let there be a source of heat S 
at temperature T^, a refrigerator i^ at a lower temperature T^, 



ART. 37] 



CARNOT'S CYCLE 



51 




and an intermediate system, the working fluid or medium M. 
The medium we may assume to be inclosed in a cylinder 
provided with a piston (Fig. IS). 

Let the medium initially in a state represented by B (Fig. 17), 
at the temperature T^ of the reservoir S^ expand adiabatically 
until its temperature falls to T^^ 
the temperature of body R. 
By this expansion the second 
state C is reached, and the 
work done by the medium is 
represented by the area B^BOC^. 
The expansion is assumed to 
proceed slowly so that the pres- 
sures on the two faces of the 
piston are sensibly equal, and 
the process is, therefore, re- 
versible. The cylinder is now 

placed in contact with R so that heat can flow from M to i2, 
and the medium is compressed. The work represented by the 
area C^CDD^ is done on the medium, and heat Q^ passes from 

the medium to the refriger- 
ator. The process is again 
assumed to be so slow as to 
be reversible. From the 
state I) the medium is now 
compressed adiabatically, 
the cylinder being removed 
from R until its tempera- 
ture again becomes T^^ that 
of the source S. During 
this third process work rep- 
resented by the area C^OBI)^ 
is done on the fluid. Finally, 
the cylinder is placed in 
contact with S and the 
fluid is allowed to expand at the constant temperature T^ 
to the initial state B. Work represented by the area 
A^ABB^ is done by the fluid during this process, and the 



J! 


\ 






illllll 


ill 


- 





Fig. 18. 



52 THE SECOND LAW OF THERMODYNAMICS [chap, iv 

temperature is kept constant by the flow of heat Q-^ from 
S to 31. 

The area ABCD inclosed by the four curves of the cycle 
represents the mechanical work gained; that is, the excess of 
work done by the medium over that done on the medium. 
Denoting this by W^ we have from the first law, 

The efficiency of the cycle is the ratio of the work gained to 
the heat supplied from the source aS'. Denoting the efficiency 
by 77, we have 

Q^ - g, _ AW 

Since all the processes of the Carnot cycle are reversible, it 
is evident that they may be traversed in reverse order. Thus 
starting from B^ the fluid is compressed isothermally from B to 
A and gives up heat Q^io S\ from A to D it expands adiabat- 
ically, from i> to C it expands at the constant temperature T^^ 
and in so doing receives heat Q^ from R ; finally it is com- 
pressed adiabatically from O to the initial state B. In this case 
the work TF" represented by area ABCD is done on the fluid M^ 
heat ^2 is taken from the refrigerator R, and the sum Q^-\- AW 
= Q^ is delivered to the source S. This ideal reversed engine 
is the basis of our modern refrigerating machines. 

38. Carnot's Principle. — The efficiency of Carnot's ideal 
engine evidently depends upon the temperatures T^ and T^ of 
the source and refrigerator, respectively. The question at once 
arises whether the efficiency depends also upon the properties 
of the substance M used as a working fluid. The answer is 
contained in Carnot's principle, namely : 

Of all engines working hetiveen the same source and the same 
refrigerator^ no engine can have an efficiency greater than that of 
a reversible engine. 

In other words, all reversible engines working between the 
same temperature limits I\ and T.^ have the same efficiency; 
that is, the efficiency is independent of the working fluid. 

The proof of Carnot's principle rests on the second law, and 
consists essentially in showing that if any engine A is more 



ART. 38] 



CARNOT'S PRINCIPLE 



53 



efficient than a reversible engine B working between the same 
temperatures, then A and B can be coupled together in such a 
way as to produce available energy without a compensating loss 
of availability. 

Suppose the two engines A and B (Fig. 19) to take equal 
quantities of heat Q^ from the source when running direct. 
Then, since by hypothesis A is the more efficient, 



and 



W.>W. 



Q/ < Q.'- 



s 


ftj 




fft 


J 


b - 




Qt 

Y 






R 



Now let engine B be run reversed. It will take heat Q^^ from 
R and deliver Q^to S, \i A and B are coupled together, A 
will run B reversed and deliver 
in addition the work Wa — Wb- 
The source is unaffected since it 
simultaneously receives heat Q^ 
and gives up heat Q^. The re- 
frigerator, however, loses the 
heat $2^ ~ Q^f^^ which is the 
equivalent of the work W^— W^ 
gained. We have, therefore, an 
arrangement by which unavail- 
able energy in the form of heat 
in the reservoir is transformed 
into mechanical work. In other 

words, by a self-acting process the available energy of the 
system of bodies aS', i2. A, and B is increased. According to 
the second law (Art. 36), such a result is impossible ; if such 
a result were possible, power in any quantity could be obtained 
from the heat stored in the atmosphere without consumption of 
fuel. 

The assumption that engine A is more efficient than the 
reversible engine B leads to a result that experience has shown 
to be impossible. We conclude, therefore, that the assumption 
is not admissible and that engine A cannot be more efficient 
than engine B. ' But if engine A is also reversible, B cannot 
be more efficient than A, and it follows that all reversible- 



Fig. 19. 



54 THE SECOND LAW OF THERMODYNAMICS [chap, iv 

engines between the same source and the same refrigerator are 
equally efficient. 

39. Determination of the Efficiency. — Since the efficiency of 
the reversible Carnot engine is independent of the properties of 
the medium and depends upon the temperatures of source and 
refrigerator only, we have 

whence ^=1 - rj = F(T,, T,) ; (2) 

that is, the quotient j~ is some function of the temperatures 

T^ and T^. The form of this function is required. 

So far, we have considered temperatures as given by a mer- 
cury or air thermometer. The different temperatures of a 
series of bodies are indicated by sets of numbers which may 
denote (1) the different lengths of a column of mercury or 
(2) the different pressures of a mass of confined gas. These 
sets may or may not precisely agree. Now there are other 
ways in which such a set of numbers may be chosen. Suppose 
we take several sources of heat /S'j, S^^ S^, •••, S^, whose tem- 
peratures are t^, t^, ^3, •••, t,^^ as defined by the mercury or gas 
scale, and let 

t^>t^>ts>"->t,. 

If we use aS'j as a source and S^ as a refrigerator, a reversible 
engine will take Q^ from S^ and deliver Q^ to S^. Since the 
bodies S^ and S^ have definite temperatures T^ and T^, what- 
ever the scale adopted, the function F(^Ty, T^ has some defi- 
nite value; therefore, from (2) the fraction -^ must have a 

definite value, and consequently §2 ^^^ one and only one value. 
If ASg is used as a source and S^ as a refrigerator, a second 

engine taking §2 fi*oiii ^2 ^^^^ ^^^® ^P ^3 ^^ ^3' ^^^ ^^ ^^* 
Starting with Q-^, we thus obtain a determinate set of values 
§2, §3, (?4, etc., which must fulfill the condition 

^1 > §2 > ^3 > - > Q.r (3) 



C^) 



ART. 39] KELVIN'S ABSOLUTE SCALE 55 

Here we have a set of nambers suitable to define a scale of 
temperature. Starting with the heat Q^ taken from the source 
aS'j, to each source there corresponds a number indicating the 
heat that would be rejected to it if it were used as a refrigerator 
in connection with S^ If we choose these numbers to define a 
new scale, then denoting the new temperatures by 

T T T ' ' . T 
we have 

whence follows 

Returning now to the quotient -^, we have at once 

hence, using this new scale, the efficiency of the Carnot engine 
is 

and the form of the function is determined. 

The scale of temperatures arrived at from the investigation 
of Carnot's cycle was first proposed by Lord Kelvin in 1848, 
and is known as the absolute scale because it is independent of 
the property of any substance. The scale is simply such that 
any two temperatures on it are proportional to the quantities 
of heat absorbed and rejected by a reversible Carnot engine 
working between these temperatures. 

If in (5) we make Q^ — ^^ 77 = 1 and T^ — 0. If we con- 
ceive a temperature lower than the zero on the absolute scale, 

T — T 

that is, if we assume a negative value for T^^ then ^ ^ — ^> 1, 

and the engine has an efficiency greater than 1, or transforms 
more heat into work than it receives from the source. Such an 
assumption is clearly inadmissible, and it follows that the zero 
of Kelvin's absolute scale is an absolute zero, and the tempera- 
ture corresponding to it is the lowest temperature conceivable. 
We are thus led to the conception of an absolute zero inde- 



56 THE SECOND LAW OF THERMODYNAMICS [chap, iv 

pendent of the properties of any particular substance. It will 
be shown subsequently that this absolute zero is precisely the 
same as that derived from the reduction in pressure of a perfect 
gas, and that the new scale coincides with that of a ther- 
mometer using a perfect gas as a fluid. 

40. Available Energy and Waste. — Carnot's ideal cycle gives 
us a means of measuring the available energy of a system and 
the waste due to an irreversible change of state. Suppose that 
a quantity of heat A§ is absorbed by the system at a tempera- 
ture T^ and that we wish to find the part of this heat that can 
possibly be transformed into work. As we have seen, no device 
can transform a larger portion of A$ into work than the ideal 
Carnot engine. If T^ is the lowest temperature that can be 

T— T 

obtained for a refrigerator, the fraction — ^-^ of A^ can be 

transformed into work by a Carnot engine, and this is, therefore, 
the availability of A$ under the given conditions. The avail- 
able part of A§ is, therefore. 



A<2^^"= A<?(l-|), 



T 

and the waste is A^ -^• 

The temperature T^ cannot be lower than that of surrounding 
objects, i.e. the atmosphere;* for even if a refrigerator could 
be found with a temperature lower than that of the atmosphere, 
it could not be maintained in that state. Hence, the tempera- 
ture of the atmosphere imposes a natural limitation on the avail- 
ability of heat in the performance of work. 

Example. If the absolute temperature of a source is 1000° F. and that of 
the atmosphere is 520^, the available energy is 

1000- 520 n/iQ -P4-V. + ^- 1 

= 0.48 of the total ener2:Y. 

1000 ^^ 

Therefore, for every 1000 B. t. u. received from the source, not more than 
480 B. t. a. can by any means whatever be transformed into work, and at 
least 520 B. t. u. must be rendered unavailable. 

* Possibly under special conditions a refrigerator whose temperature is per- 
manently below that of the atmosphere may exist ; e.g. the water of the ocean 
or of one of the great lakes. 



ART. 40] AVAILABLE ENERGY AND WASTE 57 

Let us now consider the increase of unavailable energy 
associated with certain important irreversible processes. 

(1) Conduction of Heat. — Suppose a quantity of heat Q to 
pass by conduction from a source at a temperature T^ to 
another at lower temperature T^. At the original temperature 
the available energy was 

' - 1)» 

The same quantity of heat in the second source has the avail- 
able energy 

' - Y^ 

The available energy is, therefore, decreased by the quantity 

and the unavailable energy is increased by an equal amount. 

(2) Irreversible Conversion of Work into Heat. — A common 
irreversible process is the conversion of work into heat in the 
interior of a system through the agency of friction. Examples 
are found in the flow of steam through nozzles and blades, and 
in the frictional losses due to internal whirls and eddies in 
fluids. Heat thus produced we shall denote by the symbol IT, 
reserving Q to denote heat brought into the system from outside. 

If now within the system the small quantity of heat All is 
generated while the system remains at the temperature T, the 
part of AH that is available is 



T \ T. 

where, as usual, T^^ denotes the lowest available temperature. 
Of the work J AH expended in producing the heat AH^ the 

part JAh(i-^ 

may therefore be recovered in the form of work. The re- 
mainder 

is rendered unavailable. 



58 THE SECOND LAW OF THERMODYNAMICS [chap, iv 

To obtain the total increase of unavailable energy, when the 
quantity of heat ITis generated, the temperature of the system 
varying in the meantime, we sum the element of the type just 
obtained. Thus if the temperature rises from 2\ to T^ during 
the process, we have for the total waste 

(3) Free expansion of a gas. — The waste due to free expan- 
sion, as described in Art. 35, may be determined by returning 
the gas to its initial state and observing the changes left in 
outside bodies. 

The compression indicated by B' A' (Fig. 16) requires that 
work W^ represented by area B' A' A^B^, be supplied from an 
outside body S^. Another outside body S^ must receive from 
the gas heat Q equivalent to the work W. The gas, the 
the system S^, has the same available energy as at first, being 
restored to its initial condition ; system S^^ has lost available 
energy W—JQ', and system S^ has received energy JQ of 
which only part is available. On the whole, therefore, there is 
an increase of unavailable energy. The loss of availability due 
to the original irreversible expansion of the gas (system S^ is 
repaired in this system, but an equal loss is brought about in 
systems S^ and S^. It can be shown that the waste thus in- 
curred is given by an expression of the form T^i -^' 

41. Entropy. — The expressions for the increase of unavail- 
able energy derived under various conditions are alike in hav- 
ing Tq, the lowest temperature available for a refrigerator, as a 
factor. It appears, therefore, that the unavailable energy 
changes with T^ ; the lower T^ can be taken, the smaller the 
waste and the larger the fraction of the heat supplied that can 
be transformed into work. 

The other factor in the expression must necessarily, for the 

sake of consistent units, have the form -^ or f — ^. To this 

second factor, which multiplied by T^^ gives as the product the 
increase of unavailable energy, the name increase of entropy is 



ART. 41] ENTROPY 59 

given. The change of entropy of a system is, therefore, a 
measure of the change in the unavailable energy of the system ; 
an increase of entropy involves an increase of unavailable 
energy, and vice versa. We may formally define entropy as 
follows : 

if, from any cause whatever^ the unavailable energy of a system 
is increased and if the increase he divided hy Tq^ the lowest tem- 
perature available for a cold body, the quotient is the increase of 
entropy of the system. 

This definition requires close examination to obviate possible 
misconception. The " system " spoken of may be either a 
single substance, as the medium employed in a heat motor, or 
it may be all the bodies taking part in the process. Now, ac- 
cording as we take one or the other of these viewpoints we get 
a particular notion of the significance of the term entropy. 

To illustrate this point, let us consider a simple example. 
Suppose we have a fluid medium M and a source of heat S, as 
described in connection with the Carnot engine. We may 
direct our attention either to the system illf alone or to the sys- 
tem M-\- S composed of the medium and source. Let both M 
and S be at the temperature T and suppose that at this tem- 
perature heat Q is transferred from S to M. This is the ideal 
reversible transfer assumed in the description of the Carnot 
engine. In receiving Q the system M has its available energy 

increased by QIi — ^ ] and its unavailable energy increased by 

T Q ^ 

Q~^= ^Oyr; hence by the definition just given the entropy of 

system M is increased by -^- At the same time system S has 

T 

lost the energy Q and, therefore, the unavailable energy Q~^} 

hence the entropy of S is decreased by —■ It follows that the 

change of entropy of the system M+ S is zero. As the result 
of the reversible transfer of heat from jS to M there is no 
change in the unavailable energy of the large system /S'-f-iJf and 
no change in the entropy of this system. Suppose now that sys- 
tem Jf is again at temperature T, but that system S has a higher 
temperature T' , as must be the case in any actual transfer 



60 THE SECOND LAW OF THERMODYNAMICS [chap, iv 

of heat. If now heat Q passes from S to M^ the unavail- 

T 

able energy of Mi^ increased by Q—-, as before, and the increase 

of entropy of system il[f is ^ • The system S has, however, 

T 

lost the unavailable energy §— ^, and its entropy has decreased 

Q * ^ 

by -^^' The system S+M\vds had its unavailable energy in- 
creased by the amount Q^ - Q^ = t/^ - -^\ The irre- 

versible transfer has therefore resulted in a net loss of available 
energy of this amount, and this degradation is accompanied by 

an increase of entropy — — -~ ■ The result here obtained for 

two systems may be applied to any number of systems. 

When we apply the notion of increase of entropy to the sys- 
tem composed of all the bodies involved in a process, in other 
words, an isolated system^ we are led to the conception that 
the increase of entropy measures the degradation of energy in- 
cident to the process. If we combine this notion with that 
expressed by the second law, we arrive at the following im- 
portant principles : 

1 . Any process that can proceed of itself is accompanied hy an 
increase of the entropy of the system of bodies involved in the 



2. The direction of a process^ physical or chemical^ that occurs 
of itself is such as will bring about an increase of entropy in the 
system. 

These principles lie at the foundation of the application of 
thermodynamics to chemistry. 

42. Second Definition of Entropy. — While the conception of 
entropy as the factor that measures degradation is the funda- 
mental one, we shall have occasion in succeeding chapters to 
use a much more restricted conception that has no necessary 
connection with degradation. 

Let us direct our attention to a single one of the bodies tak- 
ing part in a process, say the medium M. An increase in the 



ART. 42] SECOND DEFINITION OF ENTROPY 61 

unavailable energy of this single system involves an increase in 
the entropy of the system, but, as we have seen, degradation 
does not necessarily follow, for the increase of unavailable 
energy of M may be compensated by an equal loss in some 
other system taking part in the process. 

We now inquire by what means the unavailable energy of 
the single system under consideration can be increased. There 
are at least three ways that are suggested from the previous 
discussion of available energy (Art. 40). 

(1) If energy is added to the system in the form of heat, the 
total energy of the system is increased, and consequently the 
unavailable energy is increased. If the heat A$ is thus added 
when the temperature of the system is T, the resulting increase 
of unavailable energy is 

If, as is generally the case, the temperature rises as heat is 
added, we shall have for the increase 



C''«f="'Cf 



(2) The unavailable energy may be increased by the con- 
version of work into heat through internal friction. As shown 
in Art. 40 (2), the increase of unavailable energy from this 
cause is 

(3) If the parts of the system are not at the same tempera- 
ture, there will be an irreversible flow of heat from one part of 
the system to another, and this will increase the unavailable 
energy. We may remove this source of unavailable energy by 
assuming that the system is at all times of uniform temperature 
throughout, an assumption that is usually justifiable. 

Neglecting this third effect, we have for the increase of un- 
available energy from state 1 to state 2, 






62 THE SECOND LAW OF THERMODYNAMICS [chap, iv 

whence by definition, tlie increase of entropy is 






Now while the actual change of the system from state 1 to state 
2 may, and usually does, involve frictional effects, we can con- 
ceive of a hypothetical change in which these internal irreversi- 
ble effects are entirely absent and in which the increase of 
unavailable energy is due entirely to the addition to the system 
of heat from some external source. Denoting by Q^ the heat 
thus added, we have for the increase of entropy involved in 
this particular process the integral 

h^ T 
The important question now arises : Does the increase of en- 
tropy of the single system under consideration depend only 
upon the initial and final states or upon the path connecting 
the states? It is easily shown that the increase of entropy, 
like the increase of energy, depends upon the initial and final 
states only. For the change of energy is independent of the 
path ; therefore, the change of the unavailable part of the en- 
ergy, as determined by the constant temperature T^ and the 
temperatures T^ and T^ at the initial and final states, is also 
independent of the path ; therefore the change of entropy, 
which is the change of unavailable energy divided by T^^ is 
also independent of the path. It follows that the integral 

J~— has the same value whether taken along the path r 
(Fig. 20) or any other reversible path /. We may write, there- 

where S denotes a function of the coordinates of the system 
which may be termed the entropy of the system. We have, 
then, the following definition : 

The change of entropy of a system corresponding to a change 

of the system from state 1 to state 2 is the definite integral \ ^ -^ 

taken along any reversible path between the two states. 



ART. 43] 



INEQUALITY OF CLAUSIUS 



63 



According to this more restricted conception, the entropy of 
a system, like the energy, pressure, or temperature, is a magni- 
tude determined by the state of the system, and change of en- 
tropy has no necessary connection with degradation of energy. 

It should be noted that entropy as thus defined is like energy 
purely relative. We are never concerned with the absolute 
value of the entropy of a system in a given state ; what is 
desired is the change of entropy associated with a given change 
of state. For convenience of calculation we assume the zero 
of entropy to be the entropy of a system in some specified state. 
Thus, in dealing with vapors we assume the zero of entropy to 
be the entropy of a unit weight of liquid at 0° C. 

43. The Inequality of Clausius. — If an actual irreversible 
change be represented by the path z. Fig. 20 (assuming it to 
be possible to give such a repre- 
sentation), a correct value of the 
change cannot be obtained from 

the integral f '— ^ taken along 
the path i. For as we have seen 

T. 1 




Fig. 20. 



O 

where 2 is the increase of en- 
tropy due to the internal irre- 
versible changes. For the actual irreversible change we have, 
therefore, 

t' 



s. 



This is the inequality of Clausius. 



44. Summary. — To present the important principles of this 
chapter in concise form and in logical order the following sum- 
mary is added. 

1. Experience shows that heat energy is not completely 
transformable into mechanical work. The ratio of the energy 
that can be so transformed to the total heat energy involved 
is called the availability. 



64 THE SECOND LAW OF THERMODYNAMICS [chap, iv 

2. Experience further shows that an irreversible process 
always decreases the availability of a system. 

3. The second law of thermodynamics asserts that the avail- 
able energy of an isolated system cannot be increased by any 
process that takes place of itself. 

4. To gain a means of measuring availability the ideal Carnot 
engine is introduced. By the aid of the second law it is shown 
that no engine working between the same temperature limits 
can have an efficiency greater than the Carnot engine, and as a 
consequence, that the efficiency of this engine is a function of 
the temperature limits onl}^ 

5. By the introduction of Kelvin's absolute scale of tempera- 
ture the efficiency of the Carnot engine is found to be given by 

the fraction -J-— — ^. 

T — T 

6. Having the efficiency fraction ^ — 2, the available part 

of a given quantity of heat Q at temperature T is found to be 
Qil — ^ J and the unavailable part, Q-^- 

7. By special examples of irreversible processes it is found 
that the expression for the loss of available energy in such pro- 
cesses has the general form ^oS"^ ^^ '^ojm' 

8. The factor V-^ or f-^ which multiplied by T^^ gives the 

increase of unavailable energy is called the increase of entropy 
of the system. 

9. Two conceptions of entropy are possible : (^a) If atten- 
tion be directed to all the bodies involved in a process, the 
increase of entropy of the whole system of bodies measures the 
degradation of energy resulting from the process. (^) If at- 
tention be directed to a single body, as a medium used in a heat 
motor, the entropy of this simple system is merely a function 
of the coordinates of the system. 

10. The change of entropy of a simple system is given by 

the integral f -~ taken along any reversible path between 



ART. 45] INTERPRETATION OF THE SECOND LAW 65 

the initial and final states. The value of this integral is inde- 
pendent of the path. 

11. For an irreversible change of state the change of entropy 



is greater than j -^. 



45- Boltzmann's Interpretation of the Second Law. — A very clear insight 
into the real physical meaning of natural irreversible processes and of the 
second law of thermodynamics is afforded by the researches of Boltzraann 
and Planck. In this article it is possible to give merely a brief outline of 
Boltzraann's contribution ; for a complete exposition the reader is referred 
to Professor Klein's admirable book, The Physical Significance of Entropy.* 

According to the molecular theory, the ultimate particles of matter are 
in a state of incessant motion, the character of the motion depending upon 
the state of aggregation, — solid, liquid, or gaseous. In a gas it is assumed 
that a particle has a free path and moves along a straight line until it col- 
lides with another particle or with a restraining surface, as the wall of the 
containing vessel. To the motion of particles as to the motion of masses 
we may apply the conception of constraint or control. Thus, in the wave 
motions that characterize sound, the motion of the particles that constitute 
the mediums is in some degree controlled or ordered. The molecular 
motion that constitutes heat is, on the other hand, wholly uncontrolled and 
disordered. For any given particle of a gas all directions of motion are 
equally possible and, therefore, equally probable; and the direction of 
motion and velocity of any particle is independent of the motions of other 
particles. In a volume of gas particles will be moving in all directions 
with all possible velocities. However, because of the great number of par- 
ticles even in a small volume, the values of magnitudes that depend upon 
the molecular motion, such as pressure and temperature, remain constant 
notwithstanding the haphazard character of the molecular motion. 

According to Boltzmann, there is apparently a universal tendency 
toward the disordered motion that characterizes heat. A motion that is 
in any degree ordered or controlled tends to become disordered. Thus, as 
sound waves die out the uniform motion of the particles in the wave 
changes to disordered motion, and the energy of sound is transformed into 
heat energy. The relative motion of two bodies in contact is retarded by 
friction, and the work of overcoming friction is transformed into heat; that 
is, the constrained motion of the particles in the mass gradually changes 
to the disordered motion of heat. Since the energy of disordered molecular 
motion is necessarily less available for direction into any required channel 
than the energy of constrained or controlled motion, it follows that a change 
from a less probable state of controlled motion to a more probable state of 
disordered motion is a change from a condition of greater available energy 

*D. Van Nostrand, New York, 1910. 



66 THE SECOND LAW OF THERMODYNAMICS [chap, iv 

to a condition of less available energy. Hence, the statement of the natural 
tendency toward disordered motion is in reality a broad statement of the 
second law of thermodynamics. 

From the preceding considerations a physical interpretation of entropy 
is readily deduced. A system of itself passes from a less probable to a 
more probable state ; that is, to a state of more disordered molecular motion. 
The entropy of the system during the change must increase. Therefore, 
the entropy of the system may be associated with the probability of the 
state of the system. From the laws of probability, Planck has shown that 
the entropy is proportional to the logarithm of the probability of the state. 

The following quotations from Prof. Klein's book indicate in some degree 
the significance of this conception of entropy. 

" Growth of entropy is a passage from a somewhat regulated to a less 
regulated state." 

" Entropy is a universal measure of the disorder in the mass points of a 
system." 

" Entropy is a universal measure of the spontaneity with which a system 
acts when it is free to change." 

" Growth of entropy is a passage from a concentrated to -a distributed 
condition of energy; energy originally concentrated variously in the system 
is finally scattered uniformly in said system. In this aggregate aspect it is 
a passage from variety to uniformity." 

EXERCISES 

1. If a source of heat has an absolute temperature of 1400° F, and the 
lowest available temperature is 525° F., what fraction of the heat drawn 
from the source is available ? 

2. In a boiler 10,000 B. t. u. pass from the hot gases of the furnace, the 
temperature of which is 2500° F., through the boiler shell into water at a 
temperature of 330° F. If the lowest available temperature is 80° F., find 
the loss of available energy. 

3. Show how the result of Ex. 2 suggests the superior efficiency of the 
gas engine compared with the steam engine. 

4. Point out the loss of available energy when heat flows from steam in a 
radiator at a temperature of 225° into a room at 70°. Devise a system of 
heating that would obviate this loss. 

5. A mass of water weighing 60 lb. at a temperature of 70° F. is churned 
by a paddle wheel until the temperature rises to 120°. Find the increase of 
entropy, and the loss of available energy. Take the specific heat of water 
as 1. 

6. In the demonstration of Carnot's principle. Art. 38, assume the two 
engines A and B to do the same work W. Then show that if engine A 
is more efficient than engine B, the result is a self-acting machine that will 
convey heat from the colder to the hotter body. 



ART. 45] LITERATURE ON THE SECOND LAW 67 

REFERENCES 

Reversible and Irreversible Processes 

Planck : Treatise on Thermodynamics, Ogg's trans., 82. 

Bryan : Thermodynamics, 34, 40. 

Klein : Physical Significance of Entropy, 29. 

Chwolson : Lehrbuch der Physik 3, 443. 

Parker : Elementary Thermodynamics, 105. 

The Second Law. Entropy 

Sadi Carnot: Reflections on the Motive Power of Heat. Translated by 
Thurston. 

Clausius : Mechanical Theory of Heat. 

Rankine : Phil. Mag. (4) 4. 1852. 

Thomson : Phil. Mag. (4) 4. 1852. 

Franklin : Phys. Rev. 30, 776. 1910. 

Lorenz : Technische Warmelehre, 164. 

Chwolson: Lehrbuch der Physik 3, 485, 497. 

Bryan : Thermodynamics, 43, 57. 

Preston : Theory of Heat, 625. 

Klein : Physical Significance of Entropy. 

Magie : The Second Law of Thermodynamics (contains Carnot's " Reflec- 
tions" and the discussions of Clausius and Thomson). 

Planck: Treatise on Thermodynamics (Ogg), 86. 

Parker : Elementary Thermodynamics, 104. 



CHAPTER Y 

TEMPERATURE ENTROPY REPRESENTATION 

46. Entropy as a Coordinate. — It was shown in Art. 42 that 
the entropy of a system measured from an arbitrary zero is 
dependent only upon the state of the system ; that is, the 
entropy is a function of the coordinates of the system. It 
follows that the entropy itself may be included among the 
coordinates used to define a system. We have, therefore, five 
coordinates, namely, p^ v, jT, u, and s, that may be thus used. 
From these five, ten pairs may be selected, and the change of 
state of a system may be represented by ten different curves on 
ten different planes. Of these possible graphical representa- 
tions two are of special importance : (1) representation on the 
jt^F-plane, because the area between the curve and F-axis repre- 
sents the external work done by the system ; (2) representa- 
tion on the TyiS'-plane, because with certain restrictions the area 
under the curve represents the heat absorbed by the system 
from external sources. Graphical representations on the p V- 
plane have been considered in Art. 32. This chapter will be 
devoted chiefly to representations on the 2^S'-plane. 

From the second definition of entropy, we have 

from which relation we obtain at once the differential forms 

dS=^, (2) 

and TdS=dQ. (3) 

Let the curve AB (Fig. 21) be the path of the state-point 
projected on the TO-plane, the points A and B representing 
respectively the initial and final states. The area A^ABB^ 

68 



ART. 4' 



ISOTHERMALS AND ADIABATICS 



69 



represents the limit of the sum of terms of the type TAS; 
that is, the definite integral 

But from (3) this integral is the heat Q-^^ a,bsorbed by the 
system from external sources during the change of state. It 
follows that the area between 
the curve AB and the axis OS 
represents graphically the heat 
absorbed along the path AB. 

One most important restriction 
must, however, be observed. In 
defining entropy by means of 
equation (1) it was expressly 
stated that the change of state 
must not involve any internal 
irreversible effects. If such effects are present, the equation 
for the change of entropy is 




Fig. 21. 



s.-^\=s::'i+^' 



(4) 



where 2 denotes the increase of entropy due to internal 
processes, conduction between the parts of the system, trans- 

is the increase of entropy due to the absorption of heat from 
external bodies. From (4) it follows that in this case 



J. 



y<'S'2 



aS'j, 



whence 



dQ<TdS, (5) 

or the heat absorbed from outside is less than the area between 
the TS-Gurve and the iS'-axis. This area therefore may he taken 
as representing the heat absorbed by the system when^ and only 
when., the change of state involves no irreversible effects. Neglect 
of this restriction has led to many errors. 

47. Isothermals and Adiabatics. — If the temperature of the 
S3^stem remains constant during the change of state, the 



70 TEMPERATURE ENTROPY REPRESENTATION [chap, v 



TtS-cvLTve must be a straight line parallel to the axis OS, as 
AB (^Fig. 22). The change of entropy is simply 



Sji~ Sj_ 






T 



In this case we have merely to divide the heat added to the 
system (assuming, of course, that the change of state is revers- 
ible) by the constant tempera- 
ture T, and the quotient is the 
change of entropy. 

If the state point passes from 
B to A, that is, so as to de- 
crease the entropy, the area 
A^ABB^ represents heat re- 
jected by the system to outside 
bodies. 

For an adiabatic change of 
state, c?$ == ; hence from (1) S^ = S-^ and the adiabatic line 
on the 2W-plane, if the change of state involves no irreversible 
effects, is a straight line parallel to the ^-axis, as QD (Fig. 22). 
If the state-point moves from (7 to D, indicating a decrease of tem- 
perature, external work is done by the system, and the change 
of state is an adiabatic expansion. If the point moves upward 
from D to the change of state is an adiabatic compression. 

48. The Curve of Heating and Cooling. — From the equation 

_ ^^ 




Fig. 22. 



C = 



dl 



which defines the specific heat of a substance, we have 

dq = cdT. (1) 

Substituting this expression for dq in (1), Art. 46, we get for a 
reversible process 

f^ cdT .ox 



If the specific heat c is constant during the change of state, 
we have for the change of entropy of unit weight of the sub- 
stance 



ART. 48] THE CURVE OF HEATING AND COOLING 



71 



For the weight M, 



S-S^ = Mc\og, 






(3 a) 



If, however, c is variable, it can usually be expressed as a func- 
tion of the temperature ; that is, we can write 



whence 



_ /-^ AT^dT 



= X 



T 



(4) 



The integration can be effected when the function f(^T^ is 
knoAvn. 

Example. Let the specific heat of a substance be given by the relation 

c = a + bt^ a + b(r - 459.6) ; 
we have then 



Sl 



(a-459.66)|J-+.j'^;.r 
(a - 459.6 b) log« I^ + ^t,- t,). 



The general form of the curve that represents Eq. (3) 
is shown in Fig. 23-. This curve 
represents the ordinary pro- 
cess of heating a body or sub- 
stance, as the heating of water 
in a boiler or metal in a furnace. 
It is called by some writers the 
polytropic curve. The subtan- 
gent of the curve is constant 
and numerically equal to the 
specific heat. Thus from the 
figure we have 

JEJF = UP cot (t> 

It follows that the smaller the value of ^, the greater the slope 
of the curve. 

The isothermal and adiabatic curves (Fig. 22) mpy be con- 
sidered special cases of the heating and cooling curve. For 
the isothermal c = oo, and for the adiabatic c = 0. 




Fig. 23. 



c. 



72 TEMPERATURE ENTROPY REPRESENTATION [chap, v 




Cases may arise in which the 
slope of the T)S-Gurve is nega- 
tive, as shown in Fig. 24. In 
such cases abstraction of heat is 
accompanied by a rise in tem- 
perature or vice versa. Evidently 

the specific heat -^ must be 

negative, as is indicated geo- 
metrically by the negative sub- 
tangent. Examples will be shown in the compression of air 
in the ordinary air compressor, and in the expansion of dry 
saturated steam with the provision that it remains dry during 
the expansion. 

49. Cycle Processes. — Since any reversible process may be 
shown by a curve in TW-coordinates, it follows that a series 
of such processes forming a 
closed cycle may be repre- 
sented by a closed figure on 
the TO-plane. In Fig. 25 is 
shown such a cycle composed 
of two polytropics AB and 
2>^, an isothermal BC^ and 
two adiabatics 01) and UA. 

In any such cycle the area 
included by the cycle repre- 
sents the net heat added to 
(or abstracted from) the work- 
ing fluid during the cycle process. Assuming the cycle to be 
traversed in the clockwise sense as shown by the arrows, we 
have 

Q^ = area A^ABB^, 
Q,, = SiTe^B,BOO^, 

Qa, = area C-^BEE^, 







-. ^ 


A 






D 


E 









Fig. 25. 



C, 



ART. 50] 



THE RECTANGULAR CYCLE 



73 



Hence the excess of heat received over the heat rejected is 

Q=Q.^+Q,c+ Qa, + Q,e + Qea = A,AB B, + B,BCt\ - O^BEA^ 

= ABODE. 

If the cycle is traversed in the counterclockwise sense, we have 
evidently ^ = - area vlS (77)^. 

But from the first law, Q is the heat transformed into work ; 
hence for the direct cycle 

?.vQii ABODE =Q = AW, 
and for the reversed cycle 

area ABODE =- Q = - AW. 
This reasoning evidently holds for any number of processes, 
and therefore for a reversible 
closed cycle of any form. Thus 
for the cycle shown in Fig. 26, 
we have 

area jP= Q = AW 
or area F = — Q = — AW, 

according as the cycle is traversed 
in the clockwise or counter clock- 
wise sense. 

In later developments it will 
frequently be necessary to show cycle processes on the T/S'-plane. 

50. The Rectangular Cycle. When the curves representing 
the four processes of the Carnot cycle are transferred to the 

5%^-plane, the cycle becomes the 




Fig. 2(;. 



Ty 



-^SVH 



-^ 



B, 



Fig. 27. 



simple rectangle ABCD, Fig. 27. 
The area A^ABB^ represents the 
heat Q^ absorbed by the medium 
from the source during the iso- 
thermal expansion AB, and the area 
B^ODAy, the heat Q^ rejected to the 
refrigerator during the isothermal 
compression OD. The lines BO 
and DA represent, respectively, the 
adiabatic expansion and the adia- 
batic compression. 



74 TEMPERATURE ENTROPY REPRESENTATION [chap, v 
From the geometry of the figure, we have 

^2= ^2(^2-^1). 

whence 7) = — — = ^ ) , ,.^L-^^J^ 

as already deduced in Art. 39. 

When the cycle is traversed in the counterclockwise sense, 
the heat Q^ is received by the medium from the cold body during 
the isothermal expansion DO^ and the larger amount of heat Q^ 
is rejected to the hot body during isothermal compression BA, 
The difference Q^— Q^ = — AW represented by the cycle area 
is the work that must be done on the medium, and must there- 
fore be furnished from external sources. 

The reversed heat engine may be used either as a refrigerating 
machine or as a warming machine. In the first case the space 
to be cooled acts as the source and delivers the heat Q^ = area 
A^JDCB^ to the medium. In the second case the space to be 
warmed receives the heat Q-^ = area B^BAA^ from the medium. 

51. Internal Frictional Processes. — Referring to Art. 42, the 
increase of entropy when heat is generated in the interior of a 
system is seen to be 

s^- s,= C"^4 + C^. (1) 



^^=j.>'h;^- 



If § = 0, that is, if no heat enters the system from outside 
sources, the increase of entropy is 



S, 



^i=j*;^' (2) 

and is due entirely to the generation of heat the interior of 
the system. If it be assumed that this process is steady, so that 
the system at every instant is approximately in thermal equi- 
librium, the usual graphical representation may be applied to 
(2), and the area under the TS-q\xvyq will in this case repre- 
sent not the heat brought into the system but the heat H 
generated in the system through the agenc}^ of friction. An 
example of such a process is afforded by the flow of air or steam 



ART. 52] CYCLES WITH IRREVERSIBLE ADIABATICS 



75 




Fig. 28. 



through a nozzle. The fluid in the initial state represented by 
point A (Fig. 28) has its pressure decreased in passing along the 
nozzle, and as a result the temperature likewise falls. The 
process is adiabatic, that is, no heat 
is received from external bodies ; 
hence, if there were no internal 
friction, the drop in temperature 
would be indicated by a motion of 
the state-point along AA^ But 
work is expended in overcoming 
the friction between the fluid and 
nozzle wall. This work is neces- 
sarily transformed into heat, which 
is retained by the fluid. It follows 
that there is an increase of entropy, as indicated by the curve AB. 
From (2) the heat generated is represented by the area A^ABB^ 

52. Cycles with Irreversible Adiabatics. — In certain cases the 
closed cycle of operations of a heat motor may contain an adia- 
batic irreversible process, the irreversibility arising either from 
internal generation of heat or from the free expansion or wire- 
drawing of the working fluid. Even if it is possible to draw 
J, a TS-cvLVVG representing such 

a process, the area under that 
curve does not represent the 
heat entering the system from 
an external source. Hence 
some care is required to inter- 
pret properly the graphical 
representations of cycles with 
such irreversible parts. 

In the cycle shown in Fig. 29, 
suppose the process BO to be 
an irreversible adiabatic, the 
other parts of the cycle being reversible. Since AB is revers- 
ible, the heat absorbed in passing from ^ to jB is represented by 
the area A^ABB^ Likewise area C^ODA^ represents the heat 
rejected by the system in changing state from C to B. The 




Fig. 29. 



?6 TEMPERATURE ENTROPY REPRESENTATION [chap, v 




process IDA is adiabatic, hence Q^a = ; and by hypothesis 
Q^c — 0' The value of 2 § for tho cycle is, therefore, 

Qab + Qcd = area A^ABB^ — area C-^CDA-^^ 
= area ABUI) - area B^ECC^. 

The energy equation applies to any process, reversible or 

irreversible. Therefore for this 
cycle, as for those previously 
considered, we have 

W=JQ = J(iQ,,+ Q,,~). 

It appears, therefore, that the 
work derived is less by the area 
B^JECC^ than it would have 
been if the reversible adiabatic 
BU had been followed. 

For the reversed cycle 
(Fig. 30) w^e have as the 
work required from external sources 

W= J(Qaa + QbJ) = - area D^BAA^ + area B^BCB^ 
= -dvea. B^BCBAA^. 

Comparing this cycle with the cycle AECB having the revers- 
ible adiabatic AE. it is seen that the heat absorbed from the 
cold body is smaller by the heat represented by the area 
A^EBB^^ while the work required to drive the machine is 
greater by an equal amount. In every case the irreversible 
process results in a reduction of the useful effect. 

53. Heat Content. — Since the quantities jo, v, T^ w, and s are 
function of the state of a system only, it follows that any com- 
bination of these quantities is likewise a function of the state 
only. For example, let 

i = A{u-\-pv)', (1) 

I^AiU + pV); (la) 

E=Au-T8; (2) 

(p = Au-Ts-h Apv. (3) 

Then ^, jP, and O are magnitudes determined by the state of the 
system, and if desired, may be used along with p^ v, T as co- 
ordinates. The functions F and O are called thermodynamic 



ART. 53] 



HEAT CONTENT 



77 



potentials, and are used in certain applications of thermo- 
dynamics to physics and chemistry. The function I has use- 
ful applications in technical thermodynamics. 

To gain a physical meaning for the function I^ let us consider 
the process of heating a substance at constant pressure. If ZJp 
F^, and p^ denote the initial energy, volume, and pressure, 
respectively, and f/g, F^, and p^ the final values of the same 
coordinates, we have from the energy equation 

= A[U,^ - L\ + (p^V,^ - pj'i)}, since p^ = p^ 
= A[{U, + p.V^)~AiL\ + p,r,}] 

That is, the change in I is equal to the heat added to the sys- 
tem during a change of state at constant pressure. For this 
reason / is called the heat con- 
tent of the system at constant 
pressure^ or, more briefly, the 
"heat content." 

In some subsequent investiga- 
tions, especially tliose relating to 
the flow of fluids, it will be con- 
venient to use I and S as the in- 
dependent variables and to repre- 
sent changes of state by curves on 
the ZS'-plane. The great advantage 
of the ZS'-representation over the 

TO'-representation lies in the fact that in the former quantities 
of heat are represented by linear segments, while in the latter, 
as we have seen, they are represented by areas. A reversible 
adiabatic on the ZxS'-plane is a vertical line, sls BO (Fig. 31). 
But in this diagram segment BC represents a quantity of heat 
instead of a change of temperature. 




EXERCISES 

1. Find the change of entropy when air is heated at constant pressure 
from 70" to 200^ F., taking the specific heat as 0.24. 



78 TEMPERATURE ENTROPY REPRESENTATION [chap, v 

2. Assuming that the specific heat of water is constant, c = 1, plot on 
cross-section paper the TS -curve representing the heating of water from 
32" to 212°. 

3. Langen's formula for the specific heat of COg at constant pressure is 
Cp = 0.195 + 0.000066 t. Find, the increase of entropy when COg is heated 
at constant pressure from 500° to 2000° F. ; also the heat absorbed. 

4. A direct motor operates on a rectangular cycle between tepiperature 
limits Ti = 840° and T^ = 600° and receives from the source 200 B. t. u. per 
minute. Find the efficiency, and the work done per minute. 

5. A reversed motor, rectangular cycle, operates between temperature 
limits of 10° and 130°, and receives 600 B. t. u. per minute from the cold 
body. Find the heat rejected to the hot body, and the horsepower required 
to drive the motor. 

6. A direct motor, rectangular cycle, operating between temperatures 
T^ = 900° and T^ = 680, takes 1000 B. t. u. from a boiler. The heat rejected 
is delivered to a building for heating purposes. This direct motor drives 
a reversed motor which operates on a rectangular cycle between tempera- 
tures T^ = 460° (temperature of outside atmosphere) and T^ = 600. The 
reversed motor takes heat from the atmosphere and rejects heat to the 
building. Find the total heat delivered to the building per 1000 B. t. u. 
taken from the boiler. 

7. In the vaporization of water at atmospheric pressure, the temperature 
remains constant at 212° F., and 970.4 B. t. u. are required for the process. 
Find the increase of entropy. 

8. The expression for the energy U for a given weight of a permanent 

pV 
gas is ~- — - + Uq, where k and Uq are constants. Derive an expression for 

the heat content / of the gas. 

9. Combine the energy equation clQ = AdU + ApdV v^ith. the defining 
equation I = A{U -[■ pV) and show that dl = dQ + A vdp. 

REFERENCES 

Use of Temperature-Entropy Coordinates 

Berry : The Temperature-Entropy Diagram. 
Sankey : The Energy Diagram. 
Boulvin : The Entropy Diagram. 
Swinburne : Entropy. 

Use of Heat Content and Entropy as Coordinates 

Berry : The Temperature-Entropy Diagram, 127. 
Mollier: Zeit. des Verein. deutscher Ing. 48, 271. 
Marks and Davis : Steam Tables and Diagrams, 79. 



CHAPTER VI 

GENERAL EQUATIONS OF THERMODYNAMICS 

54. Fundamental Differentials. — The introduction of the 
entropy s and the functions i, jP, and 4> (Art. 52) permits the 
derivation of a large number of relations between various 
thermodynamic magnitudes. While the number of formulas 
that can be thus derived is almost unlimited, we shall intro- 
duce in the present chapter only those that will prove useful 
in the subsequent study of the properties of various heat media. 
In this article we shall by simple transformations express the 
differentials of u^ i^ F^ and ^ in terms of the differentials of the 
variables p, v^ T^ and s. 

We have to start with the fundamental energy equation 

dq = A{du -\- pdv}, (1) 

and for a reversible process the relation 

dq = Tds. (2) 

Combining (1) and (2), we obtain 

T 

du = — ds — pdv, (3) 

an equation that gives tt as a function of the independent varia- 
bles s and V. 

From the defining equation 

i = A(u -h pv) 
we have 

di = Adu + Ad (j)v) 

= Adu -f- Apdv + Avdp. 

Introducing the expression for Adu given by (3), we get 

di = Tds + Avdp. (4) 

79 



80 GENERAL EQUATIONS OF THERMODYNAMICS [chap, vi 

Here i is given as a function of s and p as independent 
variables. 

Likewise, from the relation 

dF = Adu - Tds - sdT; 
whence from (3) 

-dF=sdT-\-Apdv. (5) 

Finally, from the defining relation 
^ = Au 4- Apv — Ts, 
d^ = Adu + Ad(ipv) - dQTs) 

= Tds — Apdv + Apdv -{- Avdp — Tds — sdT; 
or d<^ = Avdp - sdT. (6) 

Now since the functions % ^, F, and <!> depend on the state 
only, their differentials are exact ; hence the second members 
of (3), (4), (5), and (6) are all exact differentials. 

Certain results can be deduced at once from the differential 
equations (3)-(6). For example, from (6), if a system changes 
state reversibly under constant pressure and at constant tem- 
perature, the function ^ remains constant. Again from (5), if 
a change of state occurs at constant temperature, the external 
work done is equal to the decrease of the function F. These 
results are important in the application of thermodynamics to 
chemistry. 

55. The Thermodynamic Relations. — The fact that the dif- 
ferentials in (3), (4), (5), and (6) of the last article are exact 
gives a means of deriving four important relations. In (3) 
we have u expressed as a function of the variables s and v ; 
that is, 

whence du = — ds -\ dv. 

ds dv 

Comparing this symbolic equation with (3), it appears that 

du__T du _ _ 
ds~'A' dv~ ^' 



ART. 55J THE THERMODYNAMIC RELATIONS 81 

We have now from the criterion of exactness (Art. 23), 
d fdu\ __ d fbu" 



dvKdsJ ds\dv 

that is, ii.(r) = |-(_^), 

Adv f)s 

The subscripts denote the variables held constant during the 
differentiations indicated. 

Relation (A) may be expressed in words as follows : The 
rate of increase of temperature with respect to the volume 
along an isentropic is equal to A times the rate of decrease of 
the pressure with respect to the entropy along a constant vol- 
ume curve. That is, if the reversible change of state be repre- 
sented by curves, — one on the 2V-plane, another on the ps-plane, 
— the slope of the second curve at a point representing a given 
state is — ^ times the slope of the first curve at the point that 
represents the same state. 

In (4) we have s and p as the independent variables ; and 
since di is exact, the necessary condition of exactness gives 

dp ds 

(f).=XS). <»> 

That is, the rate of increase of temperature with respect to the 
pressure in adiabatic change is A times the rate of increase of 
volume with respect to the entropy in a constant-pressure 
change. 

Since in (5) dF is an exact differential, we have 

From (6), likewise, we obtain 

The relations given by (A), (B), (C), and (D) are known 
as Maxwell's thermodynamic relations. They hold for all 



82 GENERAL EQUATIONS OF THERMODYNAMICS [chap, vi 

substances and for all reversible changes of state. For certain 
transformations the following forms, which are obtained from 

(C) and (D) by means of the relation ds = -^, are useful : 

56. General Differential Equations. — From the thermo- 
dynamic relations certain useful general equations are at once 
deduced. As in Art. 19, we may write 

according as T and v or T and p are taken as the independent 
variables. Now replacing (— ^ ] and (— ^ ) by c^ and c^, re- 
spectively, and f--^) and (— ^) by the expressions given in 
\dvjj, \dpjT 

(6^') and (i>')? these equations become, respectively, 

dq=^cJT+AT(^^dv, (I) 

dq = c^dT-AT{^^^dp. (II) 

Eliminating dT between (I) and (II), a third equation having 
p and V as the variables is obtained. Thus 

AT 



dq = 



Two other important equations may be derived from (I) and 
(II). Since from the energy equation 

du — Jdq — pdv^ 
we have iProm (I) 



du = Jc^dT-\- 



^%.-'\ 



dv ; (IV) 



ART. 56] GENERAL DIFFERENTIAL EQUATIONS 

also from the relation 

di = dq -{- Avdp, 
we have from (II) 



83 



di=CpdT-A 



T 



dv_ 
dT 



dp. 



(V) 



The general equations (I)-(y) hold for reversible changes 
of state. The partial derivatives involved may be found from 
the characteristic equation of the substance under investi- 
gation. 

As an application of (IV), we may derive expressions for the 
change of energy (a) of ' a gas that follows the Vdw pv = BT; 
(5) of a gas that obeys van der Waal's equation 



i' - s) 



p + ^,](v-b)=^BT. 



(a) From the characteristic equation pv = BT^ we have 
dp\ _B^ 



hence 



©.= 



fBT \ 

du = Jc^dT -\-{ p\dv 

= Jc.dT, 

u^ — u^ = JKc^dT 

assuming c^ to be a constant. 

(5) From van der Waal's equation, we have 



and 



whence 



H^l 



dTj, 

-P 

V — 


B 


~v-b' 

BT a 

v-b ^ v^' 



From (IV), we have, therefore, 

du = Jc^dT-{- — dv^ 
whence, assuming again that c^ is constant, 

u^-u^ = Jc^T^ - 7\) + a 



84 GENERAL EQUATIONS OF THERMODYNAMICS [chap, vi 

It appears, therefore, that if a gas follows the law pv = BT^ the 
energy is a function of the temperature only, while if it follows 
van der Waal's law, the energy depends upon the temperature 
and volume ; in other words, the gas possesses both kinetic 
and potential energy. 

/ 57. Additional Thermodynamic Formulas. — For certain in- 
vestigations of imperfect gases, especially the superheated 
vapors, certain formulas involving the specific heats Cj, and 
c^ are useful. The most important of these are (VI), (VII), 
and (VIII) following. 

Since du is an exact differential, we obtain, upon applying 
the criterion of exactness to (IV), 



dv dT 



<Si).-'] 



or 



<S)=-m-(S)r(S; 



In a similar manner, since di is exact, we have from (V) 

Equations (VI) and (VII) may be used to show the depend- 
ence of the specific heats c^ and c^ upon the pressure and vol- 
ume. For example, if a gas follows van der Waal's equation 

we find — -— = 0, whence from (VII) ( -^ ) = 0, and it follows 
dT'- ' ^ ^ \dpjj, 

that Cp does not depend upon the pressure, though it may vary 
with the temperature. Also — ^ = 0, whence it follows that c„ 

does not vary with the volume. The student may show that 
the same results follow from any characteristic equation in 
which T appears in the first degree only. If, however, we 
take the characteristic equation 



ART. 57] ADDITIONAL THERMODYNAMIC FORMULAS 85 

which applies to superheated steam, we obtain 
9% mn(n + 1) ^-, , ^ 

whence. f^-^] ^ ,4!!!<!L±1X1±M. 

\dpJr T-^^ 

Integrating this with T constant, we have 



AmnCn-i-l) /-, , a \ , j^m\ 



rpn+1 

where (^(^), an arbitrary function of T^ is the constant of inte- 
gration. In this case it is seen that e^ is a function of both T 
and p. 

An expression for Cp — c^ is obtained as follows : Writing the 
entropy s as a function of p and v, we have 

d§ = — dp -{ dv. 

dp dv 

This, combined with the familiar equation 

Adu = Tds — Apdv^ 

ds ds 

gives the equation Adu = T — dp + (T- Ap^dv. 

Since du is an exact differential, we have 

d f mds\ d f mds . V 





dv\ dp) dp\ dv ^" 


that is, 


dT ds ^ rp Sh _dT ds rp d'^S 
dv dp dvdp dp dv dpdv 


whence 


dTds dT ds _ . 
dv dp dp dv 


From the definition of specific heat, we have 




dq rp ds 



(1) 



and if we express both s and T as functions of p and v, this re- 
lation becomes 

ds n , ds J 

— dp -\ dv 

^ ^ dT . , dT -. ^^ 

dp + —-dv 
dp dv 



86 GENERAL EQUATIONS OF THERMODYNAMICS [chap, vi 
If p is constant, 0=0^ and dp — Q \ lience we have from (2) 





e 


77 ^^ 
^~ dT 

dv 




Likewise, when v is constant we have 




ds 
dp 




Combining (3) and (4), we obtain 




e^-e,= T 


' ds ds ' 
dv dp 
dT dT 
dv dp 


fdTds 

rp\dp dv 

dT 

dv 


dTds\ 
dv dp) 
dT 
dp 


Making use of ( 


1), we get 


finally 

^ _ Amdv dp 

^^ dTdT 





(3) 



(4) 



(5) 



(VIII) 



Example. For the characteristic equation j9y = BT, we have 
• dii^B dp ^B 
dT p' df v' 
Therefore, from (8), 



AB^T 



AB 



BT 



AB. 



pv pv 

That is, the difference c^ — c„ is constant even if Cp and c„ vary with the 

temperature. 

Taking Zeuner's equation for superheated steam, viz : 

pv^BT - Cp% 

V dv B dp B 
we have -^^—=1—, -^^^= — : 

dT p' dT 

BT 



whence 



c„ — c„ 



p 
AB 



nCp'^-^ + v' 
BT 



AB 



nCp"" + pv (n - 1) Cp'' + BT 

In this case, therefore, the difference Cp — c^ varies with T and p. 

By various substitutions and transformations we could add 
almost indefinitely to this list of thermodynamic formulas. 
However, the eight formulas (I)-(VIII) are sufficient for the 
investigation of nearly all problems that will arise in practice. 

It is to be observed that in the general equations of this 



ART. 58] EQUILIBRIUM 87 

chapter T must necessarily denote the temperature defined by 
the Kelvin absolute scale. The coincidence of this scale with 
the perfect gas scale will be shown in the next chapter. 

58- Equilibrium. — For irreversible processes the equations of Art. 54 
must be replaced by inequalities. Since for an irreversible process, 

dq<Tds, (1) 

Eq. (3), (4), (5), and (6) of Art. 54 become, respectively, 

Adu<Tds - Apdv, (2) 

di< Tds + Avdp, (3) 

-dF>sdT + Apdv, (4) 

d^<Avdp - sdT. (5) 

From the inequalities (4), (5), and (1) the following conclusions are at 
once apparent : 

1. If the temperature and volume of a system remain constant, then from 
(4), dF<,0. That is, dF must be negative, and any change in the system 
must result in a decrease of the function F, 

2. If the temperature and pressure remain constant, as in fusion and 
vaporization, then from (5), d^ < 0. Hence any change in the system must 
be such as to decrease the function $. 

3. If the system be isolated, ^ = 0, and from (1), ds>0. Hence in an 
isolated system any change must result in an increase of entropy. 

The conditions of equilibrium are readily deduced from these conclusions. 
Under the condition of constant T and v, change is possible so long as F 
can decrease. When F becomes a minimum, no further change is possible 
and the system is in stable equilibrium. Likewise, with T and p constant, 
stable equilibrium is attained when the function $ is a minimum. 

The functions F and <^ are evidently analogous to the potential function 
V in mechanics. A mechanical system is in a state of equilibrium when 
the potential energy is a minimum, and similarly a thermodynamic system 
is in equilibrium when either the function F or the function $ is a minimum. 
For this reason F and $ are called thermodynamic potentials. 

By the use of thermodynamic potentials, problems relating to fusion, 
vaporization, solution, chemical equilibrium, etc., are attacked and solved. 

EXERCISES 

1. From (V) derive an expression for the change of the heat content i 
when a gas following the law/>y = BT changes state. 

2. If the gas obeys van der Waal's law, find an expression for the 
change of the heat content i. 

3. Apply equations (II), (IV), and (V) to the characteristic equation 
of superheated steam, 

p(v + c) = BT-p(l + ap)^. 



88 GENERAL EQUATIONS OF THERMODYNAMICS [chap, vi 

4. Calleridar has proposed for superheated steam the equation 

pi^v-h)=BT-Cpi?^y. 

Apply (VII) to this equation and show that c is a function of p and T. 

5. Give geometrical interpretations of the thermodynamic relations 
(C) and (D). 

6. From (I) and (II) derive expressions for dq and also for -^ for a 

gas following the law pv = BT. Show that the expressions for -^ are 
integrable, while those for dq are not. 

7. Derive (VI) and (VII) by the following method : Divide both mem- 
bers of (I) and (II) by T, and knowing that -^ = ds is exact, apply the 

criterion of exactness to the resulting differentials. 

8. Deduce the following relation between the specific heats and the 
functions F and $ : 

(a)o.= -rp;(.)c..7f*. 

9. Using temperature-entropy coordinates, deduce a system of graphical 
representation for the three magnitudes Q, U^ — U^, and W that appear in 
the energy equation. 

Suggestion. Through the point representing one state draw an iso- 
dynamic, through the other point a constant volume curve. 

REFERENCES 

General Equations of Thermodynamics 

Bryan : Thermodynamics, 107. 
Preston : Theory of Heat, 637. ^ 

Chwolson : Lehrbuch der Physik 3, 466, 505. 
Buckingham: Theory of Thermodynamics, 117. 
Parker: Elementary Thermodynamics, 239. 

Equilibrium. Thermodynamic Potentials 

Planck: Treatise on Thermodynamics (Ogg), 115. 
Gibbs : Equilibrium of Heterogeneous Substances. 
Duhem : Le Potential Thermodynamique. 
Bryan : Thermodynamics, 91. 
Preston : Theory of Heat, 668. 
Chwolson : Lehrbuch der Physik 3, 519. 
Buckingham : Theory of Thermodynamics, 150. 
Parker : Elementary Thermodynamics, 325. 



CHAPTER VII 

PROPERTIES OF GASES 

59. The Permanent Gases. — The term " permanent gas " 
survives from an earlier period, when it was applied to a series 
of gaseous substances which supposedly could not by any 
means be changed into the liquid or solid state. The recent 
experimental researches of Pictet and Cailletet, of Wroblewski, 
Olszowski, and others have shown that, in this sense of the 
term, there are no permanent gases. At sufficiently low tem- 
peratures all known gases can be reduced to the liquid state. 
The following are the temperatures of liquefaction of the more 
common gases at atmospheric pressure : 

Atmospheric air - 192.2° C. 

Nitrogen - 193.1° C. 

Oxygen - 182.5° C. 

Hydrogen - 252.5° C. 

Helium - 263.9° C. 

It appears, therefore, that the so-called permanent gases are 
in reality superheated vapors far removed from temperature of 
condensation. We shall understand the term " permanent gas " 
to mean, therefore, a gas that is liquefied with difficulty and 
that obeys very closely the Boyle-Gay Lussac law. Gases that 
show considerable deviations from this law because they lie 
relatively near the condensation limit will be known as super- 
heated vapors. 

60. Experimental Laws. — The permanent gases, at the pres- 
sures usually employed, obey quite exactly the laws of Boyle 
and Charles, namely : 

1. Boyle's Law, At constant temperature, the volume of a 
given weight of gas varies inversely as the pressure. 



90 



PROPERTIES OF GASES 



[chap. VII 



2. Charles' Law. With the volume constant, the change of pres- 
sure of a gas is proportional to the change of temperature. 

By the combination of these laws the characteristic equation 
pv = BT is deduced. (See Art. 14.) In this equation T 
denotes absolute temperature on the scale of the gas ther- 
mometer, and not necessarily temperature on the Kelvin 
absolute scale. 

The classic experiment of Joule showed that permanent gases 
obey very nearly a third law, namely : 

3. Joule's Law. The intrinsic energy of a permanent gas is 
independent of the volume of the gas and depends upon the temper- 
ature only. In other words, the intrinsic energy of a gas is all 
the kinetic form. 

Joule established this law by the following experiment. Two 
vessels, a and 6, Fig. 32, connected by a tube were immersed in 
a bath of water. In one vessel air was compressed to a pres- 
sure of 22 atmospheres, the other 
^^ vessel was exhausted. The tem- 

perature of the water was taken 
by a very sensitive thermometer. 
A stopcock c in the connecting 
tube was then opened, permit- 
ting the air to rush from a to 
6, and after equilibrium was es- 
tablished the temperature of the 
No change of temperature could be 




Fig. 32. 



water was again read, 
detected. 

From the conditions of the experiment no work external to 
the vessels a and 5 was done by the gas; and since the water 
remained at the same temperature, no heat passed into the gas 
from the water. Consequently, the internal energy of the air 
was the same after the expansion into the vessel h as before. 
Now if the increase of volume had required the expenditure of 
internal work, i.e. work to force the molecules apart against 
their mutual attractions, that work must necessarily have come 
from the internal kinetic energy of the gas, and as a result the 
temperature would have been lowered. As the temperature 
remained constant, it is to be inferred that no such internal 



ART. 61] COMPARISON OF TEMPERATURE SCALES 91 

work was required. A gas has, therefore, no appreciable inter- 
nal potential energy ; its energy is entirely kinetic and depends 
upon the temperature only. 

Joule's law may be expressed symbolically by the relations : 

u=Af), 1^ = 0. 

dv 

The more accurate porous-plug experiments of Joule and 
Lord Kelvin showed that all gases deviate more or less from 
Joule's law. In the case of the so-called permanent gases, air, 
hydrogen, etc., the deviation was slight though measurable ; but 
with the gases more easily liquefied, the deviations were more 
marked. The explanation of these deviations is not difficult 
when the true nature of a gas is considered. Presumably 
the molecules of a gas act on each other with certain forces, the 
magnitudes of which depend upon the distances between the 
molecules. When the gas is highly rarefied, that is, when it is 
far removed from the liquid state, the molecular forces are van- 
ishingly small ; but when the gas is brought nearer the liquid 
state by increasing the pressure and lowering the temperature, 
the molecules are brought closer together and the molecular 
forces are no longer negligible. The gas in this state possesses 
appreciable potential energy and the deviation from Joule's 
law is considerable. 

61 . Comparison of Temperature Scales. — Joule's law furnishes 
a means of comparing the two temperature scales that' have 
been introduced : the scale of the gas thermometer and the 
Kelvin absolute scale. 

Since the intrinsic energy u is, in general, a function of T and 
v, we may write the symbolic equation 

du = ^dT-{-^dv. . (1) 

dT dv ^ -^ 



But from the general equation (IV), Art. 56, 



du = Jc^dt -f- 



^%) -^ 



dv (2) 



92 PROPERTIES OF GASES [chap, vii 

in which T denotes temperature on the Kelvin scale. Com- 
paring (1) and (2), we obtain 

For a gas that obeys Joule's law — = 0, whence from (8) 

Equation (4) is, however, precisely the equation that expresses 
Charles' law when T is taken as the absolute temperature on 
the scale of the constant volume gas thermometer. Thus, if 
the change of pressure is proportional to the change of tem- 
perature when the volume remains constant, we have, taking jOq 
as the pressure at 0° C, 



V- 
t- 


dp 
~dt~ 


- P _ 

.P-. 
T 


.P 
~ T 


(see 


Fig. 


2); 



that is. 



It follows that the value of T is the same whether taken 
on the Kelvin absolute scale or on the scale of a constant- 
volume gas thermometer, provided the gas strictly obeys the 
laws of Boyle and Joule. The fact that any actual gas, as 
air or nitrogen, does not obe}^ these laws exactly makes 
the scale of the actual gas thermometer deviate slightly from 
the scale of the ideal Kelvin thermometer. From the porous- 
plug experiments of Joule and Kelvin, Rowland has made a 
comparison between the Kelvin scale and the scale of the air 
thermometer. 

62. Numerical Value of B. — The value of the constant B for 
a given gas can be determined from the values of p, v, and T be- 
longing to some definite state. The specific weights of various 
gases at atmospheric pressure and at a temperature of 0° C. 
are given as follows : 



ART. 63] FORMS OF THE CHARACTERISTIC EQUATION 93 

Atmospheric air 0.08071 lb. per cubic foot. 

Nitrogen 0.07829 lb. per cubic foot. 

Oxygen ........ 0.08922 lb. per cubic foot. 

Hydrogen 0.00561 lb. per cubic foot. 

Carbonic acid 0.12268 lb. per cubic foot. 

A pressure of one atmosphere, 760 mm. of mercury, is 10,333 kg. 
per square millimeter= 14.6967 lb. per square inch = 2116.32 lb. 
per square foot. Taking as 491.6 the value of T on the F. 
scale corresponding to 0° C, we have for air 

^ = ^ = JL=__?1M:M_= 53.34. 
T r^T 0.08071x491.6 

In metric units the corresponding calculation gives 

B = ^^^^^ ^ 29 26 
273.1 X 1.293 

The values of B for other gases may be found in the same way 
by inserting the proper values of the specific weight 7. 

63. Forms of the Characteristic Equation. — In the character- 
istic equation as usually written, 

pv = BT, (1) 

V denotes the volume of unit weight of gas. It is convenient 
to extend the equation to apply to any weight. Letting M 
denote the weight of the gas, we have for the volume V ol M 
lb. (or kg.), V= Mv, whence instead of (1) we may write : 

pV=MBT. (2) 

This equation is useful in the solution of problems in which 
three of the four quantities, p, v, T, and M^, are given and the 
fourth is required. 

Example. Find the pressure when 0.6 lb. of air at a temperature of 
70^ F. occupies a volume of 3.5 cu. ft. ^ 
From (2) 

^ ^ ^^0-6x53.34x^0 + 459.6) ^ ^^^^^ ,^^ ^^^ ^^^^^^ ,^^^ 
= 33.63 lb. per square inch. 



94 PROPERTIES OF GASES [chap, vii 

The homogeneous form of the characteristic equation is 
advantageous in the solution of problems that involve two 
states of the gas. If (^j, F^, T^ and {p^, F^, T^ are the two 
states in question, then 

With this equation any consistent system of units may be used. 

Example. Air at a pressure of 14.7 lb. per square inch and having a 
temperature of 60° F. is compressed from a volume of 4 cu. ft. to a volume 
of 1.35 cu. ft. and the final pressure is 55 lb. per square inch. The final 
temperature is to be found. 
From (3) vp^e have 

14.7 X 4 ^ 55 X 1.35 
60 + 459.6 ^2 + 459-6' 
whence <<, = 196.5" F. 



EXERCISES 

1. Find values of B for nitrogen, oxygen, and hydrogen. 

2. Establish a relation between the density of a gas and the value of the 
constant B for that gas. 

3. Find the volume of 13 lb. of air at a pressure of 85 lb. per square inch 
and a temperature of 72° C. 

4. If the air in Ex. 3 expands to a volume of 30 cu. ft. and the final 
pressure is 20 lb. per square inch, what is the final temperature? 

5. What weight of hydrogen at atmospheric pressure and a temperature 
of 70° F. will be required to fill a balloon having a capacity of 12,000 cu. ft.? 

6. A gas tank contains 2.1 lb. of oxygen at a pressure of 120 lb. per 
square inch and at a temperature of 60° F. The pressure in the tank should 
not exceed 300 lb. per square inch and the temperature may rise to 100° F. 
Find the weight of oxygen that may safely be added to the contents of the 
tank. 

64. General Equations for Gases. — The general equations 
deduced in Chapter VI take simple forms when applied to 
perfect gases. From the characteristic equation 

pv=^BT 

we obtain by differentiation 

^ =:? (^\ =^ m 

dTJ. «' \stJp p ^ -' 



ART. 64] GENERAL EQUATIONS FOR GASES 95 

Introducing these values of the derivatives in the general 
equations (I)-(V) and (VIII), the following equations are 
obtained : 



dq- 


= c,dT + AB 


- dv, 

V 






(I 


a) 


dq = 


--c^dT-AB 


- dp, 

V 






(11 


a} 


dq = 


^B f T . ^ 


T 

' F 


#), 


(III 


a) 


du = 


--Je^dT, 








(IV 


a~) 


dl = 


= CpdT, 








(V 


a} 


(■p-<^v = 


= AB. 








(VIII 


a) 



The first two equations may be still further reduced by 
means of the characteristic equation to the forms 

dq = c^.dT-\- Apdv, (I 5) 

dq = Cj,dT-Avdp (II J) 

The ratio ^ of the two specific heats is usually denoted by 

k. The introduction of this ratio reduces (III a) to the sim- 
pler form, 

A 

dq = [_kpdv 4- vdp'] . (Ill 5) 

k—1 

Equation (IV a) simply expresses symbolically Joule's law 

that the change of energy of a gas is proportional to the change 

in temperature. Equation (15) follows independently from 

(IV a) and the energy equation ; thus 

dq = Adu + Apdv 

= c^dT + Apdv, since AJ= 1. 

EXERCISES 

1. Deduce (VIII n) from (I 6), (II 6), and the characteristic equation. 

2. Derive (V a) from (IV a) and the equation pv — BT. 

3. From (I «), (II a), and (III a) derive expressions for 

4. From (III 6) deduce the equation of the adiabatic curve in joy-coordi- 
nates. 



96 PROPERTIES OF GASES [chap, vii 

5. From (I a) derive the equation of an adiabatic in rt'-coordinates. 

6. Using the method of graphical representation explained in Art. 32, 
show a graphical representation of equation (I b). 

65. Specific Heat of Gases. — If a gas obeys the law pv = BT, 

the specific heat of the gas must be independent of the pressure 
and also independent of the volume. This principle was shown 
in Art. 57. The specific heat (c^ or c^} may, however, vary 
with the temperature, and the results of recent accurate experi- 
ments over a wide range of temperature show that such a vari- 
ation exists. As a general rule, the law of variation is 
expressed by a linear equation ; thus 

c^ = a' -{- ht. 

When the range of temperature is large, as in the internal 
combustion motor, the variation of specific heat with tempera- 
ture must be taken into account. In the greater number of 
problems that arise in the technical applications of gaseous 
media it may be assumed with sufficient accuracy that the 
specific heat has a mean constant value. 

For air the value of c^, as determined by Regnault, is 0.2375 
from 0° to 200° C. Recent experiments by Swann give the 
following values : 

0.24173 at 20° C. 

0.24301 at 100° C. 

In ordinary calculations we may take c^ = 0.24. 

The value of c^ for carbon dioxide (COg) is usually given as 
0.2012. Swann found the values 

0.20202 at 20° C, 
0.22121 at 100° C. 

The value of Cp for other gases for temperatures between 0° 
and 200° C. may be taken as follows: 

Hydrogen .... 3.4240 

Nitrogen .... 0.2438 

Oxygen .... 0.2175 

Carbon monoxide . 0.2426 

Ammonia .... 0.5106 



ART. 66] INTRINSIC ENERGY 97 

Values of the ratio k = -^ have been determined by various 

experimental methods. For air the results obtained range from 
^ = 1.39 to A; = 1.42. From the experimental evidence it seems 
probable that the true value lies between 1.40 and 1.405. In 
calculations that involve this constant, we shall take the value 
1.4 as convenient and sufficiently accurate. For air, there- 
fore, ^, = 0.24-1.4 = 0.171. 

The values of k and of e^ for other gases may be taken as 
follows ; 

k Cv 

Hydrogen 1.4 2.446 

Nitrogen 1.4 0.174 

Oxygen 1.4 0.155 

Carbon dioxide ... 1.3 0.162 
Carbon monoxide . . 1.4 0.173 
Ammonia 1.32 0.387 

If in equation (VIII a), c^ is replaced by -2, the result is the 

k 

relation 

Each of the four magnitudes c^^ k^ A, and B have been deter- 
mined experimentally, and this equation serves as a check. 

66. Intrinsic Energy. — An expression for the intrinsic 
energy of a gas is obtained by integrating (IV a). Thus 



u 



= J^cJT=Jc,T+u^, (1) 

if Cy is assumed to be constant. The constant of integration 
Uq is evidently the energy of a unit weight of gas at absolute 
zero. Since, however, we are not concerned with the absolute 
value of the energy, but the change of energy for a given 
change of state, the constant Uq drops out of consideration 
when differences are taken, and we need make no assumption 
as to its value. Hence, if (jt?^, v^, 2\) and (jt?2, ^2' ^2) ^^'^ ^^^ 
coordinate of the initial and final states, we have 

^^-u, = JcXT^-T,). (2) 



98 



PROPERTIES OF GASES 



CHAP. VII 




Fig. 33. 



This formula gives the change of energy per unit weight of 

gas. For a weight Mthe formula becomes 

U^-U^^JMcXT^-T{). (3) 

A clear understanding of the physical meaning of formula 

(2) is of such importance that it is desirable to give a second 

method of derivation, one based directly upon Joule's law. 
According to Joule's law the energy of a unit weight of gas 

is dependent on the temperature only. Hence, if ;7\, Fig. 33, 

is an isothermal, the energy 
of the gas in the state A is 
the same as in the state i); 
likewise, the energy of the 
gas at all points on the iso- 
thermal T^ is the same. It 
follows that the change of 
energy in passing from tem- 
perature T-^ to temperature T^ 
is the same, whether the path 
is-1^, AC, 01 DU. 
Since the energy is directly proportional to the temperature, 

the change of energy is directly proportional to the change of 

temperature. Hence 

in which a denotes a proportionality-factor. To determine the 
factor a, we choose some particular path between the isother- 
mals T^ and T^ (Fig. 33). As we have seen, if this constant 
is established for one path it holds good for every other path. 
The most convenient path for this purpose is a constant volume 
line, as AC. The heat required for a rise in temperature from 
^1 toy, is q,,=^eXT,-T{). 

Since in the constant volume change, the external work is zero, 
we have from the general energy equation 

Comparing these equations, we have 

u^-u^ = Jc^(T^-T^^l 
therefore, the constant a in (4) is Jc^. 



ART. 67] HEAT CONTENT 99 

A formula for the change of energy in terms of p and V may 
be derived from (3). Multiplying and dividing the second 
member by B^ 

- k-1 • <-'^^ 

In (5) V2 and V\ denote the final and initial volumes, respec- 
tively, of the weight of gas under consideration; consequently 
it is not necessary to find the weight iJf in order to calculate the 
change of energy. It is to be noted, however, that in using 
(5) pressures must be taken in pounds per square foot. 

Example. Find the change of energy when 8.2 cu. ft. of air having a 
pressure of 20 lb. per square inch is compressed to a pressure of 55 lb. per 
square inch and a v&lurae of 3.72 cu. ft. 

Using the value k = 1.40, 

_ ^r ^ 144 ^ 55x3.72-20x8.2 ^ ^^^ 

^ ^ 0.40 

67. Heat Content. — The change in heat content correspond- 
ing to change of state of a gas is readily derived from the 
general equation (Y a). 

Thus, i = I c^dT= c^T+ i, (1) 

and i,-i, = c^iT,-T,). (2) 

Introducing the factor AB in the second member of (2), 



C. 



(P2^2-Pl^D 



k 
= Aj^—jCp^v^-p^v^}. (3) 

For a weight of gas M, (2) and (3) become, respectively, 

I,-I, = Mo,iT^-T,), (4) 



100 PROPERTIES OF GASES [chap, vii 

and ^2-^i = ^]^(P2^"2-Pi^i)' (5) 

68. Entropy. — Expressions for the change of entropy are 
easily derived from the general equations (I «), (II a), and 
(Ilia}. Dividing both members of these equations by T, we 
have 

, do dT . T^dv ' ^-.x 

ds = -^=c,^ + AB-^, (1) 

ds = '-^=c/-^-AB'-l. (2) 

ds = Cp \-c^-^. (3) 



Hence for a change of state from (p^, v^, T^ to (p^, v^, T^, 

82-Si=«.l0g.||+^i?l0g.^ (4) 

C^=o„\og,^-AB\og,^ (5) 

^1 Pi 



= ., log. ;;?+.. log, f. (6) 

^1 P\ 

These formulae give the change of entropy per unit v^-^eight 
of gas. For any other weight M^ the change of entropy is 
ilf (§2 — Si). Equations (4), (5), and (6) are in reality identi- 
cal. Each can be derived from either of the other two by 
means of the relations pv — BT^ c^ — c^ = AB. In the solution 
of a problem, the equation should be chosen that leads most 
directly to the desired result. 

EXERCISES 

1. From (4), (5), and (6) deduce expressions for the change of entropy 
corresponding to the following changes of state : (a) isothermal, (h) at con- 
stant volume, (c) at constant pressure. 

2. By making s^— s•^^ = in (4), (5), and (6), deduce relations between 
Tand v, T and j9, and/> and v for an adiabatic change of state, 

69. Constant Volume and Constant Pressure Changes. — In 

heating a gas at constant volume the external work is zero. 
Hence 

Q = A(iU^- U,-) = McXT^- T,-). (1) 



ART. 69] 



CONSTANT PRESSURE CHANGES 



101 



The change of entropy is 

= Jf^,logA (2) 

Pi 

When the gas is heated at constant pressure, the external 

work is 



W,,=piV,-V,). 



(3) 



The increase of energy is, as in all cases, given by the 
relation 



rr _ rr ^ pV^j-pVi ^ "^12 
^2 ^1- yt_l -^^Tl* 



The heat added is, therefore, given by the relation 



k 



k-1 

or using the relation jt? F^= MBT^ 



K^2-^i); 



Cl2 = 



A<^p-<^v) 



which might have been written directly. 
The increase of entropy is 

Equations (2) and (7) may 
also be derived directly from 
the general equations for en- 
tropy, Art. 68. 

The changes of state just 
considered are represented on 
the ^iS'-plane by curves of the 
general form shown in Fig. 34. 
The curve AB^ which rep- 
resents the constant volume O 
change, is steeper than the 



(4) 



(5) 



MBi T^ - T{) = Mc,i T^ - Ti), (6) 



O) 




102 PROPERTIES OF GASES [chap, vii 

curve AC, which represents heating at constant pressure. This 
follows from the inequality 

that is, area A^ABB^ < area A^ACC^. 

70. Isothermal Change of State. — li T is made constant in 
the equation ^F= MBT^ the resulting equation 

pY= p^V^ = constant (1) 

is the equation of the isothermal curve in p F^coordinates. This 
curve is an equilateral hyperbola. The external work for a 
change from state 1 to state 2 is given by the general formula 



TF.o = 



12 



= f Vf: (2) 

Jv 



Using (1) to eliminate j9, we have 

= MBT\og,^. (3) 

For the change of energy, 

U,-U, = JMc, iT,-T,) = 0; (4) 

hence rr 

Qn = AW,.^ = Ap,V,log,-^, (5) 



and 



•^2 - 'S'l = % = ^*^log, ^ = ABMlog, ^. (6) 



Since in isothermal expansion the work done is wholly sup- 
plied by the heat absorbed from external sources, it follows that 
if the expansion is continued indefinitely, the work that may be 
obtained is infinite. This is also shown by (3), thus : 






71. Adiabatic Change of State. —To derive the j9v-equation of 
an adiabatic change of state, we may use the general difPeren- 



ART. 71] ADIABATIC CHANGE OF STATE 103 

tial equation containing p and v as variables. The most con- 
venient form of this equation is (III a), 

dq = (ydp + kpdv). (1) 

K — i 

During an adiabatic process no heat is supplied to or ab- 
stracted from the system ; hence in (1) dq = 0, and therefore 

vdp -f- kpdv = 0. (2) 

Separating the variables, 

dp kdv _ r. 

p V 

whence loggjo -j- k log^ v = log (7, 

or pv^ = O. (3) 

The relation between temperature and volume or between 
temperature and pressure is readily derived by combining (3) 
and the general equation pv — BT. Thus from 

pv^ = (7, 
pv = RT, 
we get by the elimination of jt?, 

v'-^ = — ; 
BT' 

that is, Tv^-^=2 const. (4) 

Similarly, by the elimination of v, we obtain 

T>k 

^k-\ — -P rpk . 

^ ~ o ' 

k-\ 

that is, =^— = const. (5) 

If we choose some initial state, p-^^ Vj, 2\, the constants in 
(4) and (5) are determined, and the equations may be written 
in the homogeneous forms 



T fp\ " 
T, \pj 



(6) 

k-i 

O) 



104 PROPERTIES OF GASES [chap, vii 

Since in an adiabatic change the heat Q is zero, the energy 
equation gives 

whence using the general expression for the change of energy, 






(8) 



By means of the equation 

the final volume F^ may be eliminated from (8). The result- 
ing equation is 

Example. An air compressor compresses adiabatically 1.2 cu. ft. of free 
air {i.e. air at atmospheric pressure, 14,7 lb. per square inch) to a pressure 
of 70 lb. per square inch. Find the work of compression; also the final 
temperature if the initial temperature is 60° F. 

For the final volume, we have 



Fs = 1.2 C—Y' = 0.3936 cu. ft. 



The work of compression is 

PiVi-p^V2 ^ 144(14.7 X 1.2 - 70 X 0.3936) ^ _ o.gg ,, ,, 
k-1 0.4 *" * 

The initial temperature being 60° + 459.6° = 519.6° absolute, we have for 
the final temperature 

T2 = 519.6 /"-^Y' = 811.6 abs., 

whence t2 = 352° F. 

72. Polytropic Change of State. — The changes of state con- 
sidered in the preceding sections are special cases of the more 
general change of state defined by the equation 

pV''=: const. (1) 



ART. 72] POLYTROPIC CHANGE OF STATE 105 

By giving n special values we get the constant volume, constant 
pressure, and other familiar changes of state. Thus : 

fo^?^ = 0, pv^ = const., i.e. jt? = const, 

for 71=00, p^v = const., v = const, 
for n = Ij pv = const., isothermal, 

for n = k, pv'' = const., adiabatic. 

The curve on the p F'-plane that represents Eq. (1) is called 
by Zeuner the poly tropic curve. 

By combining (1) with the characteristic equation jo V=MBT, 
as in Art. 71, the following relations are readily derived 

n-\ n-\ 

For the external work done by a gas expanding according to 
the law pV^ =p-^ V^" = const., 

from the volume F^ to the volume Fg, we have 

V 



=i-i^a i_/ J 



1 — n 

The change of energy, as in every change of state, is 



(3) 



U,- U.^P^fll^. (4) 

rC — J. 

Hence, from the energy equation, we have for the heat absorbed 
by the gas during expansion 

K — 1 1 — n 

Comparing (3), (4), and (5) we note that the common factor 
(P2 ^2~ Pi ^i) occurs in the second member of each expression. 



u^-u^ 


1- 


-n 


w 


k- 


-1 


JQ 


k- 


1 
- n 


JQ _ 


_^- 


- n 



106 PROPERTIES OF GASES [chap, vii 

Hence, dropping out this factor, we may write the following 
useful relations : 

W k-1 ^3^ 

These may be combined in the one expression 

W: U^- Ui:JQ = k-l:l-n:k-n. (9) 

Example. Let a gas expand according to the law 

p V^-^ = const. 
Taking k = lA, we have 

W : Uz- Ui'.JQ = OA : - 0.2 :0.2 = 2 : - 1:1; 

that is, the external work is double the equivalent of the heat absorbed by 
the gas and also double the decrease of energy. 

73. Specific Heat in Polytropic Changes. — From Eq. (5), 
Art. 72, an expression for Q-^^ in terms of the initial and final 
temperatures of the gas may be readily derived. Since 

p^ Fi = MBT^, and p^ V^ = MBT^, 
(5) becomes 

^ _MAB k-n.rp rp^ 



But 



AB ^^ 



k-1 



hence, Qn = ^^J^QT^- 2\). (1) 

We have, in general, 

Q^^= MciT^- T{) . (2) 

where c denotes the specific heat for the change of state under 
consideration. Comparing (1) and (2), it appears that 

k — n ^o\ 

c = c^- . (3) 

1 — n 

Hence, for the polytropic change of state, the specific heat is con- 



ART. 73] SPECIFIC HEAT IN POLYTROPIC CHANGES 



107 



tfp, and the 




Fig. 35. 



stant (assuming c^ to be constant) and its value depends on the 
value of n in the equation p V^ = const. 

It is instructive to observe from (3) the variation of ^ as w 
is given different values. For ti = 0, c = kc 
change of state is repre- 
sented by the constant- 
pressure line aa^ Fig. 35, 
36. For n = l^ e = oo, and 
the change of state is iso- 
thermal (line 6). li n='k^ 
then c = 0, and the ex- 
pansion is adiabatic (line 
d). For values of n lying 
between 1 and Ar, the value 
of c as given by (3) is 
evidently negative ; that 
is, for any curve lying 
between the isothermal h 

and adiabatic d, rise of temperature accompanies abstraction of 
heat, and vice versa. This is shown by the curve c. 

It will be observed that 

in passing through the 

region between curves a 

and J, n increases from 

to 1 and c increases from c^, 

to oo ; then as n keeps on 

increasing from ^ to 1, c 

changes sign at curve h by 

passing through go, and 

increases from — oo to 0. 

As n increases from n=^k 

to n= -{- oo, c increases 

^ from c=0 to c— c^; and 

for n= CO, the constant 

volume case, c becomes c^. 

As we proceed further, n changes sign and increases from 

— oo to 0, the value of c in the meantime increasing from 

Cv to Cj,. 




Fig. 36. 



108 PROPERTIES OF GASES [chap, vii 

74. Determination of n. — It is frequently desirable in experi- 
mental investigation to fit a curve determined experimentally 
— as, for example, the compression curve of the indicator 
diagram of the air compressor — by a theoretical curve having 
the general equation pV^= c. To find the value of the 
exponent n we assume two points on the curve and measure to 
any convenient scale jo-^, p2-> ^v ^^^ ^t Then since 

we have n = ^ ^ ; ^ — , ?} - (1) 

log V^- log V^ 

Example. In a test of an air compressor the following data were 
determined from the indicator diagram : 

At the beginning of compression, p^ = 14.5 lb. per square inch. 

Fj = 2.56 cu. ft. 
At the end of compression, p2 = 68.7 lb. per square inch. 

Fa = 0.77 cu. ft. 

Assuming that the compression follows the law pV* = const., we have 
for the value of the exponent 

^^ log 68.7 -log 14.5 ^-^30 
log 2.56 - log 0.77 

The work of compression is 

^ ^ P^V^-PiVi _ 144(68.7x 0.77- 15.5x2.56) ^ _ ^^^^ ^ j^ 
^^ 1-n -0.32 ' * 

The increase of intrinsic energy is 

U, - U, ^a-Il^llL^ = 144(68.7x0.77-14.5x2.56) ^ ^gg^ ^^_^^. 

tC — J. U.4 

and the heat absorbed is 

^ 5680 - 7100 1 00 T3 X 

Q12 ;^ = - 1.83 B. t. u. 

The negative sign indicates that heat is given up by the air during com- 
pression ; this is always the case with a water-jacketed cylinder. 

If the initial temperature of the air is 60° F., or 519.6° absolute, the final 
temperature is 



r, = 519.6(M5y-^^. 763.2, 
\0.77/ ' 

whence h = 303.6° F. 



ART. 74] DETERMINATION OF n 109 

EXERCISES 

1. A curve whose equation is pV^ = C is passed through the points 
p^ = 40, Fi = 6 and p^ = 16, V^ = 12.5. Find the value of n. 

2. Air changes state according to the law pV^ = C. Find the value of 
n for which the decrease of energy is one half of the external work ; also the 
value of n for which the heat abstracted is one third of the increase of energy. 

3. If 32,000 ft.-lb. are expended in compressing air according to the 
law p F^-^^ = const., find the heat abstracted, and the change of energy. 

4. In heating air at constant pressure 35 B. t. u. are absorbed. Find 
the increase of energy and the external work. 

5. A mass of air at a pressure of 60 lb. per square inch absolute has a 
volume of 12 cu. ft. The air expands to a volume of 20 cu. ft. Find the 
external work and change of energy : (a) when the expansion is isothermal ; 
(h) when the expansion is adiabatic ; (c) when the air expands at constant 
pressure. 

6. If the initial temperature of the air in Ex. 5 is 62° F., what is the 
weight? Find the heat added and the change of entropy for each of the 
three cases. 

7. Find the specific heat of air when expanding according to the law 
pv^-^ = const. If during the expansion the temperature falls from 90° F. to 

— 10° F., what is the change of entropy? 

8. Find the latent heat of expansion of air at atmospheric pressure and 
at a temperature of 32° F. 

9. The volume of a fire balloon is 120 cu. ft. The air inside has a 
temperature of 280° F., and the temperature of the surrounding air is 70° F. 
Find the weight required to prevent the balloon from ascending, including 
the weight of the balloon itself. 

10. A tank having a volume of 35 cu. ft. contains air compressed to 
90 lb. per square inch absolute. The temperature is 70° F. Some of the 
air is permitted to escape, and the pressure in the tank is then found to be 
63 lb. per square inch and the temperature 67° F. What volume will be 
occupied by the air removed from the tank at atmospheric pressure and at 
70° F. ? 

11. Air in expanding isothermally at a temperature of 130° F. absorbs 
35 B. t. u. Find the heat that must be absorbed by the same weight of air 
at constant pressure to give the same change of entropy. 

12. Air in the initial state has a volume of 8 cu. ft. at a pressure of 
75 lb. per square inch. In the final state the volume is 18 cu. ft. and the 
pressure is 38 lb. per square inch. Find: (a) the change of energy; (h) the 
change in the heat content ; (c) the change of entropy. 

13. Find the work required to compress 30 cu. ft. of free air to a pressure 
of 65 lb. per square inch, gauge according to the lawjoy^-^ = const. Find the 
heat abstracted during compression. 



no PROPERTIES OF GASES [chap, vii 

14. Deduce the relation Cp — c^ = AB from first principles without 
recourse to general equations. 

Suggestion. Let one pound of gas be heated through the temperature 
range Tg ~ ^\ (^) ^* constant volume, (h) at constant pressure. Find an 
expression for the excess of heat required for the second case and then 
make use of the energy equation. 

15. Suppose the specific heat of a gas to be given by the linear relation 
Cy = a -{- bt. Deduce relations between p, v, and T for an adiabatio change. 

Suggestion. Use the general equation dq = c^dT + Apdv and the char- 
acteristic equation pv = BT. 

REFERENCES 

Characteristic Equation of Gases. Deviation from the 
Boyle-Gay Lussac Law 

Zeimer : Technical Thermodynamics (Klein) 1, 93. 

Preston : Theory of Heat, 403. 

Barus: The Laws of Gases. N".Y. 1899. (Contains the researches of 

Boyle and Amagat.) 
Regnault : Relation des Experiences 1. 
Weyrauch : Grundriss der Warme-Theorie 1, 124, 127. 

The Porous-plug Experiment. The Absolute Scale of 
Temperature 

Thomson and Joule : Phil. Trans. 143, 357 (1853) ; 144, 321 (1854) ; 152, 

579 (1862). 
Rose-Innes: Phil. Mag. (6) 15, 301. 1908. 
Callendar: Phil. Mag. (6) 5, 48. 1903. 
Olszewski : Phil. Mag. (6) 3, 535. 1902. 

Buckingham : Bui. of the Bureau of Standards 3, 237. 1908. 
Preston : Theory of Heat, 695. 
Bryan : Thermodynamics, 128. 
Chwolson : Lehrbuch der Physik 3, 546. 

Specific Heat of Gases 

Regnault : Relation des Experiences 2, 303. 

Swann: Proc. Royal Soc. 82 A, 147. 1909. 

Zeuner : Technical Thermodynamics (Klein) 1, 116. 

Chwolson : Lehrbuch der Physik 3, 226. 

Preston : Theory of Heat, 339, 243. 

Weyrauch : Grundriss der Warme-Theorie 1, 146. 

General Equations 

Zeuner : Technical Thermodynamics 1, 122. 
Weyrauch : Grundriss der Warme-Theorie 1, 152. 
Bryan: Thermodynamics, 116. 



CHAPTER YIII 

GASEOUS MIXTURES AND COMPOUNDS. COMBUSTION 

75. Preliminary Statement. — In the preceding chapter we 
discussed the properties of simple gases with the implied 
assumption that chemical action was excluded. For many 
technical applications a knowledge of such properties is suffi- 
cient for the consideration of all questions that arise. On the 
other hand, investigations of the greatest importance, those 
relating to internal combustion motor, have to deal with 
gaseous substances that enter into chemical combination and 
(after combustion) with mixtures of inert gases. In the 
present chapter, therefore, we shall consider some of the pro- 
perties of gaseous compounds as dependent on chemical com- 
position, and also the properties of mixtures of gases. 

76. Atomic and Molecular Weights. — Let E^^ JE^, etc. denote 
different chemical elements and a^, a^^ etc. their corresponding 
atomic weights. Then if n-^^ n^, etc. denote the number of atoms 
of ^j, ^2' ^^^' entering into a molecule of a given combination, 
the molecular weight of the compound is 



m 



Tij^a-^ + n^a^ + ••• etc. = ^na. (1) 



For the elements that enter into subsequent discussions the 
atomic weights (referred to the value 16 for oxygen) are as 
follows : 

Approximate 
Exact Value Integral Value 

Hydrogen 1.008 1 

Oxygen 16.000 16 

Nitrogen 14.040 14 

Carbon 12.000 12 

Sulphur 32.060 32 

111 



1:12 GASEOUS MIXTURES AND COMPOUNDS [chap, viii 

The approximate integral values are sufficiently exact for 
practical purposes, in view of unavoidable errors in experi- 
mental results. 

Using these values, we have as the molecular weights of cer- 
tain important substances the following : 



Water 


H.O 


m=2x 14-1x16 = 18 


Carbon monoxide CO 


1 X 12 H- 1 X 16 = 28 


Carbon dioxide 


00, 


1x12 + 2x16 = 44 


Ammonia 


NH3 


lxl4 + 3x 1 = 17 


Methane 


CH4 


lxl2 + 4x 1 = 16 


Nitrogen 


^2 


2 X 14 = 28 


Hydrogen 


H. 


2x1 = 2 



The composition by weight of a compound is readily deter- 
mined from the value of n, a, and m. Thus in a unit weight 
(pound) of compound there is 

'^h^l lb. of element K, 
m 

^^ lb. of element U,, etc. 
m 

For example, COg is composed by weight of ^^ carbon and 
II oxygen, NHg is composed by weight of ^|- nitrogen and ^j 
hydrogen. 

77. Relations between Gas Constants. — If in the character- 
istic equation pv = BT, which holds approximately for any 

gaseous substance (mixture or compound), we replace v by - 
we have '^ 

^7 = 1- (1) 

Here 7 denotes the weight of unit volume of the gas. From 
this relation it is seen that for a chosen standard pressure and 
temperature, for example, atmospheric pressure and 0°C., the 
product By is the same for all gases. But since the specific 
weight 7 of a gas is directly proportional to the molecular 
weight w, it follows that the product Bm is likewise the same 



ART. 77] RELATIONS BETWEEN GAS CONSTANTS 113 

for all gases. Denoting this product Bm by i?, we have for 
the characteristic equation of any gas 

pv^^T, (2) 

m 

From (1) we obtain the relation 

B = Bm = P^^; (3) 

hence the numerical value of B can be found when the values 
of m and 7 are accurately known for any one gas. From Mor- 
ley's accurate experiments, we have for oxygen 7 = 0.089222 lb. 
per cubic foot at atmospheric pressure and 32° F. ; and for 
oxygen m = 32. Inserting these numerical values in (3), we 
obtain 

P^ 2116.3x32 ^-.r.. 
0.089222 X 491.6 

The constant B is called the universal gas constant. From it 
the characteristic constant B of any gas can be determined at 
once from the molecular weight. Thus for carbonic acid we 
have 

^ = 1^=35.09. 
44 

From the general formula 

c^-c, = AB (4) 

for the difference between the specific heats of a gas, we have 

^ ^AB^ 1544 1 ^ 1.9855 .^. 

^ ^ m 777.64 m m 

This relation gives a ready method of calculating one specific 

heat from the other when the molecular weight m is known. 

Thus for CO2, c^-c,= ^-^^^^ = 0.0451, and if t?^ = 0.2020, we 

44 

have c, = 0.2020 - 0.0451 = 0.1569. 

It is convenient to express the specific weight 7 and the 

specific volume v of a gas in terms of the molecular weight m. 

These constants are referred to standard conditions, namely, 

atmospheric pressure and a temperature of 32° F. From (3) 

we have 7 = -^ w, (6) 

BT 



114 



GASEOUS MIXTURES AND COMPOUNDS [chap, viii 



whence inserting the numerical values, jt? = 2116.3, 72 = 1544, 
^=491.6, 

7 = 0.002788 m. (7) 

For the normal specific volume, we have 



or 



V = 



V = 



7 p m 

3 58.65 
m 



(8) 



(9) 



From the preceding relations, the following values are readily 
found for the constants of certain gases. 



Gas 


Chemical 
Symbol 


Molecular 
Weight 

m 


Character- 
istic 
Constant 


Difference 

of Specific 

Heats 

Cp-C^ 


Weight per 
cubic feet at 

32° F. and 
Atmospheric 

Pressure 


Volume per 
pound 
at 32° F. 
and Atmos- 
pheric 
Pressure 


Oxygen .... 
Hydrogen . . . 
Nitrogen . . . 
Carbon dioxide . 
Carbon monoxide 
Methane . . . 
Ethylene . . . 
Air 


O2 
CO2 

CO 
CH, 
C,H, 


32 

2.016 
28.08 
44 
28 

16.032 
28.032 


48.249 
765.86 
54.985 
35.09 
55.142 
96.314 
55.079 
53.34 


0.0621 
0.9849 
0.0707 
0.0451 
0.0709 
0.1238 
0.0708 
0.0686 


0.089222 

0.005621 

0.07829 

0.12268 

0.078028 

0.04470 

0.078036 

0.08071 


11.208 
177.9 
12.773 
8.151 
12.81 
22.37 
12.794 
12.39 



78. Mixtures of Gases. Dalton's Law. — A mixture of several 
gases that have no chemical action on each other obeys very 
closely the following law first stated by Dalton : 

The pressure of the gaseous mixture upon the walls of the con- 
taining vessel is the sum of the pressures that the constituent gases 
wovld exert if each occupied the vessel separately. 

Like Boyle's law, Dalton's law is obeyed strictly by mix- 
tures of ideal perfect gases only. Mixtures of actual gases show 
deviations from the law, these being greater with gases most 
easily liquefied. For the purpose of technical thermodynamics, 
however, it is permissible to assume the validity of Dalton's 
law even in the case of a mixture of vapors. 

Let T^ denote the volume of a given mixture, ilffj, J^, iHfg, . . . 
the weights of the constituent gases, and B^^ B^, B^, . . . the 



ART. 78] MIXTURES OF GASES. DALTON'S LAW 115 

constants for those constituents ; then the partial pressures of 
the constituents, that is, the pressures they would exert sepa- 
rately if occupying the volume V, are : 

M^B^T _ M^B,T _M^B^T 

Pi— Y ' i^2 — pr" ' i^3 — jr ' '" v^J 

According to Dalton's law the pressure p of the mixture is 

i' = /'i-rft + ft+ - =^(.M^B^ + 3L,B^+M^B^+ ■■■). (2) 

Furthermore, if Jf is the weight of the mixture, 

M=M-^-[-M^-hM^-{- ••• =^Mi. (3) 

Let us now introduce a magnitude B^ defined by the equation 

MB^ = M,B, + M^B^ + M,B, + • • ; (4) 

then (2) takes the form 

pV=MB„T. (5) 

The constant B^ may be regarded as the characteristic con- 
stant of the mixture. It is obtained from (4), which may be 
written in the more convenient form 

The partial pressures may readily be expressed in terms of 
the pressure of the mixture. Thus combining (1) and (5), 
we obtain 

Pi_Mi^i ^_^2^2 etc m 

Example. A fuel gas has the composition by weight given below. The 
value of the constant Bm for this gas is found as indicated by the following 
arrangement : 

CONBTITUENTR WEIGHT B MB 

CO2 0.04 35.09 1.4036 

CO 0.27 55.142 14.8883 

H2 0.067 765.86 51.3126 

N2 0.585 54.985 32.1652 

CH4 0.033 96.314 3.1784 

C2H4 0.005 55.079 0.2754 

1.000 103.24 



116 GASEOUS MIXTURES AND COMPOUNDS [chap, viii 

Since ikf = 1 and 2 MB = 103.24, we have 

Brr, = 103.24. 
The apparent molecular weight of the mixture is 

1544 -,. (.n. 

m = = 14.96, 

103.24 

and the weight per cubic foot under standard conditions is, therefore, 

y = 0.002788 X 14.96 = 0.0417 lb. 

79. Volume Relations. — Let V^, 1^, Fg, •••, denote the vol- 
ume that would be occupied at pressure p and temperature T 
by several gaseous constituents ; then if B^, B^, B^, - • •, denote 
the characteristic constants of these gases, we have 

pV^ = M^B^T, pV^ = M,B,T, pV^ = M^B^T, ••. (1) 

If now the gases be mixed, keeping the same pressure and 
temperature,' the mixture will occupy the volume 

V= V,+ V^+ V,+ ..., (2) 

and its weight will be necessarily 

M=M^ + M^-\-M^+ '- (3) 

Taking B^ as the characteristic constant of the mixture, we 
have 

pV=MB„T. (4) 

Comparing (1) and (4), we obtain the relations 

V MBj V-MB^'" ^^^ 

It will be seen that the volume ratios given by (5) are equal 
to the pressure ratios given by (7) of Art. 78. 

If 7 denotes the weight of a unit volume (1 cu. ft.) of gas, 
then 

7 = ^ = f. (6) 

For the several constituents of a mixture, we have, therefore, 

Jfi = 7iF;, M^ = 7,F2, M, = 73 F3, -, (7) 

and for the mixture 

7» = f=-^(7iFi + 72r, + 73n+-)- (8) 



ART. 80] COMBUSTION: FUELS 117 

Similarly, we have for the specific volume of the mixture 

Since 7 = 0.002788 m = km (see Art. 77), we have from (7) 

and M= M^^- M^+ ... = ^Sm^Ti. 

Therefore, -^ = vr^ ' ^ = ^ ^ t^ , • • • (10) 

If further we denote by m^ the product ^^7^, we have from (8) 

^m = yS^iF;. (11) 

The constant w^ we ma}^ regard as the apparent molecular 
weight of the mixture, and from it we may determine the con- 
stants B^^ Cp— <?^, 7, and v of the mixture. 

Equations (10) and (11) are useful in the investigation of a 
mixture when the composition hy volume is given. The follow- 
ing example shows the method of procedure. 

Example. A producer gas has the composition by volume given below. 
Required the composition by weight and the constants of the mixture. 

Constituents Volume V m mV — 

H2 0.08 2 0.16 0.006 

CO 0.22 28 6.16 0.2308 

CH4 0.024 16 0.384 0.0144 

CO2 0.066 44 2.904 0.1088 

N2 0.61 28 17.08 0.64 

1.000 26.688 1.000 

According to (10) the last column gives the composition by weight. The 
constant m,n is 26.688; hence we have 

B^ = iMl = 57.85. y = 0.002788 x 26.688 = 0.07441. 

26.688 ^ 

1 Q9,^^ 358.65 io AA 

c„-c, = :^^^^ = 0.0744. ^ = K^-x^ = 13.44. 

p 26.688 26.688 

80. Combustion : Fuels. — The elements that chiefly combine 
with oxygen to produce reactions characterized by the evolution 
of heat are hydrogen and carbon. Cornj)ounds that are made 



118 



GASEOUS MIXTURES AND COMPOUNDS [chap, viii 



up largely of these elements are fuels ; for example, methane 
CH^, benzol CgHg, alcohol CgHgO. The product of complete 
combustion of hydrogen is HgO, water ; that of complete com- 
bustion of carbon is CO2, carbon dioxide. Sulphur is a possible 
constituent of fuels, and the product of combustion is SO2, sul- 
phur dioxide. 

Chemical reactions are, in general, characterized by the evo- 
lution or absorption of heat. The union of a combustible with 
oxj^gen is accompanied by the evolution of a considerable 
quantity of heat, and the heat evolved by the combustion of a 
unit weight of the combustible is called the heating value of 
the combustible. The heating value of hydrogen alone or car- 
bon alone must be determined by experiment, but the heating 
value of a compound of C and H may be calculated, at least 
approximately. 

Hydrogen and compounds containing hydrogen have two 
heating values, called respectively the higher and the lower. 
This arises from the fact that the product HgO may be either 
water or steam. If the temperature after combustion is above 
212°, the product exists as vapor, and the heat necessary to 
keep it in the vapor form is not set free ; hence, we have the 
lower heating value. If, however, the vapor condenses, the 
heat of vaporization is recovered, and we have the higher heating 
value. 

The heating values of various fuels are given in the follow- 
ing table. 







B. T. u. PEK Pound 


B. T. xj. PER Cubic 
Foot at 82° F. 

Low 




High 


Low 


Hydrogen .... 

Carbon 

Carbon monoxide . 
Methane .... 
Ethylene .... 
Acetylene .... 


Ho 

c; 

CO 
CH, 
C,H, 
C,H, 


62100 
14600 
4380 
23842 
21429 
21429 


52230 
14600 
4380 
21385 
20025 
20673 


294 

342 

956 

1563 

1499 



The heating value of a fuel mixture is determined from the 
heating values of the separate constituents. Denoting by ilff^, 



IT 


MH 


52230 


313.38 


4380 


1010.9 


21385 


307.94 



ART. 81] AIR REQUIRED FOR COMBUSTION 119 

Mc^^ •••, the weights of the constituents, by jHj, H^, H^^ •••, the 
corresponding heating values per pound, and by H^ the heat- 
ing value of the mixture, we have 

whence n„ = ^^^. (1) 

By a similar procedure the heating value per cubic foot may 
be obtained when the composition by volume is given. 

Example. Required the lower heating value of the producer gas de- 
scribed in the example of Art. 79. 

For the heating value per pound we have 

M 

H2 0.006 

CO 0.2308 

CH4 0.0144 

2 M^= 1632.2 

Since M = 1, we have Hra='^ MH = 1632.2 B. t. u. per lb. 

The heating value per cubic foot (at 32° F. and atmospheric pressure) is 
evidently the product 

Hrr^y = 1632.2 x 0.07441 = 121.5 B. t. u. 

Or from the composition, we have 

F 

H2 '. 0.08 

CO 0.22 

CH4 0.024 

121.7 B. t. u. per cu. ft. 

The difference in the two results is due to approximations in the calculation, 
and is of no importance. 

81. Air required for Combustion and Products of Combustion. 

— The oxygen required for the complete combustion of a given 
fuel is determined from the equation of the reaction. For 
example, the combustion of methane, CH^, is given by the 
equation 

CH4 + 2 O2 = CO2 + 2 H2O ; 

that is, two molecules of oxygen combine with one molecule of 
methane and the resulting products are one molecule of COg 



H 

294 


VH 

23.52 


342 


75.24 


956 


22.94 



120 GASEOUS MIXTURES AND COMPOUNDS [chap, viii 

and two molecules of HgO. Since by Avogadro's law the 
volumes are proportional to the numbers of molecules enter- 
ing into the equation, we may also read the preceding chemical 
equation as follows : two volumes of oxygen combine with one 
volume of CH^, producing one volume of COg and two volumes 
of H2O. 

Taking the molecular weights of the four gases into con- 
sideration, we may write the equation 

16 + 2 X 32 = 44 + 2 X 18. 

From this it appears that one pound of CH^ requires for com- 
plete combustion ^4. = 4 lb, of oxygen and the products are 
41 = 2.75 lb. of CO2 and ff = 2.25 lb. of HgO. 

Since oxygen is 23 per cent of air by weight, the weight of 
air required for the complete combustion of one pound of CH^ 

4 
is = 17.4 lb. The volume of air required for the burnino^ 

0.23 ^ ^ 

of one cubic foot of CH^ is — ^ = 9.52 cu. ft. 

We may generalize the process illustrated by the preceding 
example as follows : 

Let the gaseous fuel have the composition C^^H^^O^g, and let 
fflj, «2i <^3 denote the atomic weights of C, H, and O, respectively. 
Then the molecular weight of the fuel in question is 

m = a-(n^ + ^2^2 + %%• 

The equation of the reaction may be written 

C„,H.,0„, + xO^ = yCO^ + zhJo, 

where x^ y^ and z indicate the number of molecules of the 
respective substances. Comparing the two members of the 
equation, we find 

2 = i ^2, 

^3 -}- 2 2; = 2 «/ + 2, 

whence x — J(2 y -\-z — n^ =n^-[-^n^— ^n^. 

From these results we have the volume of oxygen and the 
volumes of the products of combustion CO2 and H2O. Thus 



ART. 8]] AIR REQUIRED FOR COMBUSTION 121 

for alcohol CgHgO, a: = 2 + | — | = 3, «/ = 2, and 2; = 3, showing 
that for the combustion of one cubic foot of alcohol vapor, 
3 cu. ft. of oxygen are required, and the resulting products are 
2 cu. ft. of CO2 and 3 cu. ft. of HgO. 

To get the relations between the weights of the substances 
under consideration we must introduce the molecular weights 
in the reaction equation. Thus we obtain 

m + 2 a^x = yQa^ + 2 a^ -\- z(2 a^^ ag), 

from which follow the ratios : 

, _ weight of oxygen _ 2 a^x _ a^ .^ . _ . . 

weight 01 luel m m 

, weiorht of CO9 y(a-. + 2 ao) n. . , o n 

y' = — 7^ ^ = ^^ ^ ^ = -^ (a. + 2 a«) ; 

weight ot fuel mm 



weight of HgO _ g(2 a^ + g^) _ n^ .^ . 

weight of fuel ~ m ~ 2m^ ^^ ^^' 



If we make use of the integral values of the atomic weights, 
namely, ^^ = 12, a^ = 1, ^g = 16, we have for the complete com- 
bustion of one pound of the combustible : 

x' = oxygen required = — (2 ^^ + '^n^ — n^ lb. ; 

y' = COo produced = 44 -i lb. ; 

m 

z' = HoO produced = 9 ^ lb. 
m 

Taking alcohol, CgHgO, for example, we have 

71-^^ = 2^ n^ = 6^ n^ = l, m= 2 X 12 + 6 X 1 -{-1Q = 46, whence 

x' = 11(2 X 2 + 1 X 6 - 1) = 2.087 ; 

y'=^^i^ = 1.913; 
'^46 

2^ = 5-^ = 1.174. 
46 

The weight of air required per pound of alcohol is 

1^=9.075 lb. 
0.23 



122 GASEOUS MIXTURES AND COMPOUNDS [chap, viir 

and the weight of nitrogen appearing among the products of 
combustion is, therefore, 9.075 - 2.087 = 6.988 lb. 

If a gaseous fuel is a mixture of several combustible con- 
stituents, the values of x^ ^ y\ and z' may be found for the indi- 
vidual constituents separately. Then if M^^ M^^ M^^ •••, are the 
weights of the constituents respectively, we have 

""" M '^~ M '^~ M ' 

Example. For the producer gas heretofore investigated, we have the 
following values : 





M 


93' 


y' 


s' 


Mx' 


My' 


Mz' 


H. 


0.006 


8 





9 


0.048 





0.054 


CO 


0.2308 


0.571 


1.571 





0.1318 


0.3626 





CH4 


0.0144 


4 


2.75 


2.25 


0.0576 


0.0396 


0.0324 


CO2 


0.1086 





1 








0.1088 





^2 


0.64 





















1.00 0.2374 0.511 0.0864 

One pound of the gas requires 0.2374 lb. of oxygen for complete combustion. 
The weight of air required is, therefore, 0.2374 - 0.23 = 1.032 lb., and this 
air brings with it 1.032 - 0.2374 = 0.7946 lb. of nitrogen. We have then the 
following balance : 



CONSTITITENTS 








PEODTTCTa 






Fuel gas 


1.00 


lb. 






CO2 




0.511 lb. 




Air 


1.032 






H2O 




0.0864 






2.032 lb. 






N2 


0.64 + 0.7946 


= 1.4346 


















2.032 lb. 




Taking the composition 


by 


volume, the 


following results are found 


: 


V 




X 




y 


z 


FCB 


Vy 


Vz 


H^ 0.08 




0.5 







1 


0.04 





0.08 


CO 0.22 




0.5 




1 





0.11 


0.22 





CH^ 0.024 




2 




1 


2 


0.048 


0.024 


0.048 


CO2 0.066 









1 








0.066 





N2 0.61 
























1.00 












0.198 


0.31 


0.128 



Since 0.198 cu. ft. of oxygen is required per cubic foot of gas, the volume of 
air required is 0.198 -^ 0.21 = 0.943 cu. ft., and the volume of nitrogen corre- 
sponding is 0.943 - 0.198 = 0.745 cu. ft., which is added to the 0.61 cu. ft. in 
the fuel gas. The volume of gas and air before combustion is 1 + 0.943 — 
1.943 cu. ft., and the volume of the products is 0.31 + 0.128 + 0.61 + 0.745 
= 1.793 cu. ft. Hence there is a contraction in volume of 7.8 per cent. 



ART. 82] SPECIFIC HEAT OF GASEOUS PRODUCTS 123 

82. Specific Heat of Gaseous Products. — In deducing the 
special equations for gases we assumed that the specific heat 
of any gas remains constant at all pressures and temperatures. 
In many technical applications this assumption is sufficiently 
near the truth and is justified by the simplicity of the analysis 
based upon it ; but when a very wide range of temperature is 
encountered, as in the case of the internal combustion motor, the 
assumption of constant specific heat is no longer permissible. 

The gaseous products that come under consideration may be 
separated into two classes. (1) The simple or diatomic gases, 
as nitrogen, oxygen, air, etc. ; (2) the compounds, like carbon 
dioxide (CO2) and steam (HgO), which may be regarded as 
superheated vapors rather than as gases. For the products in 
the first group, the law pv = BT holds quite exactly, and, there- 
fore (see Art. 57), the specific heat must be independent of the 
pressure, but may vary with the temperature. The substances 
in the second group, which, are comparatively near the liquid 
state, do not follow the gas law closely, and for these the 
specific heat may vary with the pressure as well as with the 
temperature. The character of the variation of the specific heat 
of steam is shown in Fig. 71, Art. 133. At the lower tempera- 
tures the specific heat increases with the pressure, but as the tem- 
perature rises the influence of the pressure becomes negligible 
and the specific heat rises with the temperature. It is probable 
that the specific heat of COg varies in somewhat the same manner. 

Experiments on the specific heats of various gases show that 
in general the specific heat rises with the temperature, and that 
the law governing the variation is expressed sufficiently well 
by the simple linear equation 

e = a -\- ht. 

The formulas, as usually stated, give molecular specific heats, 
the molecular specific heat being numerically equal to the 
thermal capacity of a weight of the substance expressed by the 
molecular weight. Thus, since the molecular weight of carbon 
monoxide (CO) is 28, the molecular specific heat of CO is 
numerically equal to the thermal capacity of 28 pounds of CO. 
We may denote molecular specific heat by the product mc. It 



124 GASEOUS MIXTURES AND COMPOUNDS [chap, viii 

is an interesting fact that while the specific heats of simple 
gases are quite different, the molecular specific heats are sub- 
stantially identical. 

The results of Langen's experiments are given by the follow- 
ing formulas, in which t denotes temperature in degrees C. 

For all simple gases 

m(?, = 4.8 + 0.0012 ^. ' (1) 

For carbon dioxide 

mc^ = 6.7 + 0.0052 t. (2) 

For water vapor 

mc^ = 5.9 + 0.0043 t. (3) 

Dividing by the appropriate value of the molecular weight m, 
the heat capacity of a gas per unit weight is readily found. 
Thus for oxygen m = 32, and from (1) we have 

,?^ = 0.15 + 0.0000375^; 

for COg, m = 44, and from (2) we obtain 

tf, = 0.1523 + 0.0001182^. 

Formulas (1), (2), and (3) give molecular specific heats at 
constant volume. From the relation m((?p— c^)= 1.9855 (see 
Art. 77), we have approximately mCp = mc^ + 2. Therefore, 
from the preceding equations we obtain corresponding equa- 
tions for Cp, namely : 

'mcp = 6.8 + 0.0012^; (4) 

mc?p = 8.7 + 0.0052^; (5) 

wcp= 7.9 + 0.0043^. (6) 

For temperatures F. the preceding formulas become respec- 
tively: 

1. For simple gases 

<?, = -(4.77 + 0.000667 



m 

= 1(4.48 + 0.000667 7^) 
m 

= 1(6.75 + 0.000667 
m 

= 1(6.46 + 0.000667 ^) 
m 



(7) 



(8) 



ART. 83] SPECIFIC HEAT OF A GASEOUS MIXTURE 125 

2. For carbon dioxide 

(?^= 0.15 + 0.000066^ 1 

= 0.12 + 0.000066^1 

Cp = 0.195 + 0.000066^ 1 
= 0.165 + 0.000066 T) ' 

3. For superheated water vapor 

c^= 0.324 + 0.000133^ 1 

= 0.263 + 0.000133 t\ 
(?p= 0.435 + 0.000133^ 1 

= 0.374 + 0.000133 T\ 



(9) 
(10) 

(11) 
(12) 



83. Specific Heat of a Gaseous Mixture. — Let M^, M^, -" 
respectively, denote tlie weights of the constituents of a mix- 
ture and (?^^, e^^^ •••, the corresponding specific heats. It is 
assumed that for a given temperature rise each constituent 
requires the same quantity of heat when mixed with other 
constituents as it would if separated from them. Hence, the 
heat Q required for a temperature change T^ — T^ is 

Q = M.c^X T; - T,y + M^c^XT^ - T,) + .... 
But we have also 

where M= M^-\- M^-h •••, and c^ denotes the specific heat of 
the mixture. Combining these expressions, we obtain 

Likewise, ^^^ ~M^' ^^^ 

Example. Find the specific heat Cy of a mixture of 1 lb. of the pro- 
ducer gas described in the example of Art. 79 and 1.25 lb. of air, which is 
about 20 per cent in excess of the air required for complete combustion. 
Find also the specific heat Cy of the products of combustion. 

Of the 1.25 lb. of air furnished 0.2625 lb. is oxygen and 0.9875 lb. is 



126 GASEOUS MIXTURES AND COMPOUNDS [chap, viii 

nitrogen. Adding this nitrogen to the 0.64 lb. in the gas, the total nitrogen 
is 1.6275 lb. We have then 

0.006 





M 


m 


H2. . . . 


... 0.006 


9, 


CO . . . 


. . . 0.2308 


28 


CH4 . . . 


. . . 0.0144 


16 


C02 . . . 


. . . 0.1088 




N2. . . . 


. . . 1.6275 


98 


0. . . . . 


. . . 0.2625 


82 



(4.48 + 0.000667 T) 

(4.48 + 0.000667 T) 

^ (4.48 + 0.000667 T) 
16 

0.1088(0.12 + 0.000066 T) 

1.6275 



2 
0.2308 

28 
0.0144 



28 

0.2625 

32 



(4.48 + 0.000667 T) 
(4.48 + 0.000667 T) 



^M = 2.25 :S,Mc, = 0.3646 + 0.0000597 T 

Hence, e. = 0-^646 + 0.0000597 T ^ ,^^^0 + 0.0000265 T. 
2.25 

For the products of combustion, we have (see Art. 81) 

If Mcv 

CO2 0.511 0.511(0.12 + 0.000066 7^) 

H2O 0.0864 0.0864(0.263 + 0.000133 T) 

.627 
28 

32 



N2 1.6275 L6275 ^^_^g ^ 0.000667 T) 

28 

O2 0.0251 2:2251 (4 43^ 0.000667 T) 



Therefore, 



^M = 2.25 -^Mc, = 0.34394 + 0.00008451 T 

0.34394 + 0.00008451 T 



2.25 
= 0.1529 + 0.00003756 T. 

84. Adiabatic Changes with Varying Specific Heats. — When 
the specific heat of a gas is taken as a function of temperature, 
thus 

c^ = a^ + hT, 

the simple relations derived in Art. 71 no longer apply. We 
have as before, however, 

dq= c^dT-\- Apdv, (1) 

Cp- c^ = AB, (2) 

^2-^1 = -^^'.(^2-^1)- (3) 



ART. 85] TEMPERATURE OF COMBUSTION 127 

For an adiabatic change dq = ; hence from (1), we have 

c^dT= — Apdv^ 

or (a + hT^dT=-ABT^. C4) 

V 

From (4) we obtain upon integration 

alog^^+biT^-T,)=ABlog^'!l. (5) 



T, 



From the characteristic equation pv — BT^ we have —J =o_2^ 
therefore (5) becomes 

or «log.|? + H2'2-^i) = (^-B + «)log.^- (6) 

Finally, if in (5) we substitute for — its equivalent ^— ^, we 
obtain 

whence AB log,^^ =(^ + ^^) log^ ^ + j(2; - T^). (T) 

For the external work of adiabatic expansion, we have 

To 



= JMQia-^hT)dT 



Equations (5), (6), and (7) are readily applied when the 
initial and final temperatures are given. When, however, 
the final temperature is required, the equation in T is tran- 
scendental and its solution requires a process of successive 
approximations. The illustrative example of the following 
article shows the method of procedure. 

85. Temperature of Combustion. — A close analysis of the pro- 
cess of burning a fuel gas under given conditions involves com- 
plicated equations, especially when the specific heat is taken as 
variable. The temperature and pressure at the end of the pro- 



128 GASEOUS MIXTURES AND COMPOUNDS [chap, viii 

cess are the results usually desired, and these may be found, at 
least approximately, by a simple method. 

Let T^ denote the temperature of the gaseous mixture at the 
beginning of combustion and T^ the desired final temperature ; 
H the lower heating value of the fuel per pound, and M the 
combined weight of one pound of fuel and of the air furnished 
for combustion (Jf is evidently also the weight of the products 
of combustion). It is assumed that the combustion is complete, 
and that the heat ^is all expended in raising the temperature 
of the products from T^ to T^. As a matter of fact, the com- 
position of the mixture during the combustion process is con- 
tinually changing, but as the specific heat changes but little, it 
is considered permissible to base the calculation on the final 
products alone. We have then 

ir=M('^\a + bT)dT, (1) 

where a-\-hTis the expression for the variable specific heat of 
the products. From (1) we obtain upon integration 

<T,-T,) + ^~CT,^-T,^)=^ (2) 

from which T^ may be calculated. 

Example. The mixture of producer gas and air in the example of 
Art. 83 is compressed adiabatically from an initial pressure of 14.7 lb. per 
square inch to a pressure of 150 lb. per square inch absolute. The initial 
temperature is 530° absolute. The mixture is then burned at constant 
volume and the products of combustion expand adiabatically to the initial 
volume. Required the temperature and pressure after compression, after 
combustion, and after expansion. Also the work of compression and the 
work of expansion. 

The characteristic constants of the fuel mixture and of the mixture of 
the products, respectively, are first required. For the fuel mixture we have 

M B MB 

H2 0.006 765.86 4.59516 

CO ...... 0.2308 55.142 12.72678 

CH4 0.0144 96.314 1.38692 

CO2 0.1088 35.09 3.81780 

:N'2 1.6275 54.985 89.50808 

O2 0.2625 48.249 12.65936 

2.25 124.694 

B = 124.694 - 2.25 = 55.42 : AB = 0.07127. 



ART. 85] TEMPERATURE OF COMBUSTION 129 

For the mixture of products, we obtain B = 51.81 ; AB = 0.06662. 
For the fuel mixture, the expression for the specific heat is 

c, = 0.1620 + 0.0000265 T. 

AVe have, therefore, from (7), Art. 84 

0.23327 loge ^ = 0.07127 loge — - 0.0000265(7^2 - Tx). 

To solve this equation for To let us assume as a first approximation 
2\ - Ti = 500. Then 

1 Z^^ 0-16555 -0.01325^ 
^ Ti 0.23327 

and ^ = 1.921. 

Therefore, T.^ = 1.921 x 530 = 1018.1, 

and T2- Ti= 488.1. 

As a second approximation, we assume T-2 — Ti= 490. We obtain 



T2 _ 0.16555 - 0.01299 
Ti ~ 0.23327 

T2 



0.6540, 



= 1.9233, T2 = 1.9233 x 530 = 1019.3, 
Ti 

T2- Ti = 489.3. 

As the assumed value of T2 — Ti is so nearly attained, we may take the 
value T2 = 1020 as sufficiently exact. 

The ratio of initial and final volumes is now readily found from the 

relation £iFi.= £iZH. 

Ti T2 

Thus, i^ = a.I^ = lMxl020^0.1887. 

Vi p2 7^1 150 530 

For the external work required to compress one pound of the mixture, we 
have 

W = / r*^" (0.1620 + 0.0000265 T)dT = 69550 ft.-lb. 

If Tz, denotes the temperature after combustion, we have from (2), taking 
c„ = 0.1529 + 0.00003756 T for the products of combustion, 

0.1529(r3 - 1020)+ ^^^^^^^^(7;2 _ 10202) =}^^^-^ 

whence T^ = 3963°. 

To find the pressure 7)3, we must take account of the change of composi- 
tion during combustion. For the initial state, p2V= 55.42 T2, at the end of 
combustion 7)3 r= 51.81 73. Hence, we have 

n . 5L81 ^ 150 ^ 3963 ^ 5i^ ^ 544.8 lb. per sq. in. 
^' ^-T, 55.42 1020 55.42 ^ , 

K 



130 



GASEOUS MIXTURES AND COMPOUNDS [chap, vin 



For the adiabatic expansion, the ratio of volumes is the same as for the 
adiabatic compression. That is, —=0.1887. 

From (5) Art. 84, we have 

a log, p = h{T, - r,) + AB loge ^% 
which may be written in the form 

Inserting the known values ^4^ = 0.06662, a = 0.1529, & = 0.00003756, 



T^ = 3963, ^= 0.1887, we get 

log T^ = 3.70517 - 0.0001067 T^. 

This equation may be solved 
graphically, as shown in Fig. 37. As 
the value of T4, evidently lies betw^een 
2500° and 3000° we plot the curves 

y = log T, 
and y = 3.70517 - 0.0001067 T^ 

between these limits. The intersec- 
tion gives the desired value, 
7^4 = 2647°. 
The external work of expansion is 

W= J rj(0.1529 + 0.00003756 T)dT 
== 283,600 ft.-lb. 




EXERCISES 



The following are the compositions by volume of two gases, one a rich 
natural gas, the other a blast furnace gas : 



Natural Gas (Indiana) 

H2 ...... . 0.02 

CO 0.007 

CH, 0.931 

C2H4 0.005 

O2 0.004 



CO2 



0.003 
0.030 



Blast Furnace Gas 

Hg 0,05 

CO 0.27 

CH4 0.015 

O2 0.085 

Ng a58_ 

1.00 



1.00 



ART. 85] EXERCISES 131 

Work the following examples for each of these gases: 

1. Find the composition by weight. 

2. Find the heating value : 

(a) per cubic foot mider standard conditions; 
(6) per pound. 

3. Calculate the constants ^m, y, y, and Cj(, — c„. 

4. Find the volume of air required for the combustion of one cubic 
foot. 

5. Find the weight of air required for the combustion of one pound. 

6. Find the products of combustion, by w^eight. 

7. Find the specific heat c„ of a mixture of the gas with air, the weight 
of air being 15 per cent in excess of that required for complete combustion. 

8. Find c„ for the products of combustion, assuming that 15 per cent 
excess of air is used. 

9. Find the constants £'", y, and v of the mixture of Ex. 7 ; also of the 
products of combustion. 

10. The mixture of Ex. 7 is compressed adiabatically from a pressure of 
14.7 lb. per square inch to a pressure of 120 lb. per square inch in the 
case of the natural gas and to a pressure of 175 lb. per square inch in the 
case of the blast furnace gas. The initial temperature in each case is 80° F, 
Find the temperature at the end of compression in each case. 

11. Find the work of adiabatic compression. 

12. Find the ratio of initial to fi.nal volume. 

13. If at the end of adiabatic compression the mixture is ignited and 
burns at constant volume, find the temperature at the end of the process, 
assuming that no heat is lost by radiation. 

14. After combustion the products expand adiabatically to the initial 
volume. Calculate the final temperatures. 

15. Find the work of adiabatic expansion. 

16. Assume that the adiabatic compression follows the law joF**= const. 
Find the values of n. Find also the values of n for the adiabatic expansion. 

REFERENCES 

Gas Mixtures 

Preston : Theory of Heat, 350. 

Bryan: Thermodynamics, 121. 

Zeuner : Technical Thermodynamics (Klein) 1, 107. 

Weyrauch : Grundriss der Warme-Theorie 1, 137, 140. 



132 GASEOUS MIXTURES AND COMPOUNDS [chap, viii 

Fuels. Combustion. Heating Values 

Levin : Modern Gas Engine and Gas Producer, 80. 

Carpenter and Diederichs : Internal Combustion Engines, 129. 

Zeuner : Technical Thermodynamics 1, 405, 416. 

VVeyrauch: Grundriss der Warme-Theorie, 216, 255. 

Jones : The Gas Engine, 293. 

Poole : The Calorific Power of Fuels. 

In the field of thermochemistry reference may be made to tlie exten- 
sive researches of Favre and Silbermann, Berthelot, and J. Thomsen. For 
tables of heating values see Landolt and Bornstein : Physik.-chemische 
Tabellen. 

Variable Specific Heat of Gases 

Mallard and Le Chatelier : Ann ales des Mines 4. 1883. 

Vieille : Comptes rendus 96, 1358. 1883. 

Langen : Zeit. des Verein. deutsch. Ing. 47, 622. 1903. 

Haber : Thermodynamics of Technical Gas Reactions, 208. 

Clerk : Gas, Petrol, and Oil Engines, 341, 361. 

Zeuner : Technical Thermodynamics 1, 146. 

Carpenter and Diederichs : Internal Combustion Engines, 220. 

Thermodynamics of Combustion 

Zeuner : Technical Thermodynamics 1, 423, 428. 

Lorenz : Technische Warmelehre, 392. 

Stodola : Zeit. des Verein. deutsch Ing. 42, 1045, 1086. 1898. 



CHAPTER IX 

TECHNICAL APPLICATIONS. GASEOUS MEDIA 

86. Cycle Processes. — In any heat motor, heat is conveyed 
from the source of supply to the motor by some medium, which 
thus simply acts as a vehicle or carrier. In practically all 
cases the medium is in the liquid or gaseous state, though a 
motor with a solid medium is easily conceivable. The perform- 
ance of work is brought about by a change in the specific 
volume of the medium due to the heat received from the source. 
By a proper arrangement of working cylinder and movable pis- 
ton this change of volume is utilized in overcoming external 
resistances. (In the steam turbine another principle is em- 
ployed.) The medium must pass through a series of changes 
of state and return eventually to its initial state, the series of 
changes thus forming a closed cycle. To use a crude illustra- 
tion, the medium taking its load of heat from the source at high 
temperature, delivering that heat to the working cylinder and 
to the cold body (condenser) and returning to the source for 
another supply may be compared with an elevator taking 
freight from an upper story to a lower level and returning 
empty for another load. 

Where the medium is expensive it is used over and over, and 
thus passes through a true closed cycle. Examples are seen in 
the ammonia refrigerating machine and in the engines and 
boilers of ocean steamers, in which fresh water must be used. 
In such cases we may speak of the motor as a closed motor. 
If the medium, on the other hand, is inexpensive or available in 
large quantities, as air or water, open motors are quite generally 
used. In these the working fluid is discharged into the atmos- 
phere and a fresh supply is taken from the source of supply. 
Even in this case the medium may pass through a closed cycle, 

133 



134 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix 

but all the changes of state are not completed in the organs of 
the motor. 

In this chapter we shall take up the analysis of several cycles 
that are of importance in the technical applications of gaseous 
media. In general, we shall assume ideal conditions, which 
cannot be attained in actual heat motors. However, the con- 
clusions deduced from the analysis of such ideal cycles are 
usually valid for the modified actual cycles ; furthermore, the 
ideal cycle furnishes a standard by which to measure the effi- 
ciency of the actual cycle. 

87. The Carnot Cycle. — Although the Carnot cycle is of no 
practical importance, it possesses the greatest interest from a 
theoretical point of view. Hence an analysis of it is included. 

Referring to Fig. 18, the heat absorbed from the source dur- 
ing the isothermal expansion AB is given by the equation 

Qa, = Ap,V,l0g,^, (1) 

and the heat rejected to the refrigerator is 

Q,a = Ap,Klog,^. (2) 

The heat transformed into work is, therefore, 

A W= Qao + Q,a = A(pa K log. ^ - Pc K loge ^). (3) 

Since in the state A the temperature is :7\, we have 

p^K = MBT,, (4) 

and likewise p,V, = MBT^. (5) 

Furthermore, for the adiabatic BO we have the relation 

VJ T, . ^^> 

From (6) and (7) we have 

}^=Y^. (8) 



vJ r; 

and for the adiabatic DA the relation 



ART. 88] CONDITIONS OF MAXIMUM EFFICIENCY 



135 



Introducing in (3) the results given by (4), (5), and (8), we 
obtain 

K 



AW=MB(T,-T,)log,-^ 



whence 



AW 



MB(iT,-T,)log, 



V = 



Qo 



MBT.log,^ 






T. 



(9) 



(10) 



T, 



dY- 



To 



This expression for the efficiency is identical with that deduced 
from the Kelvin absolute scale of temperature. We have in 
Eq. (10) a proof, therefore, that the Kelvin absolute scale coin- 
cides with the perfect gas scale. 

88. Conditions of Maximum Efficiency. — On the T/S'-plane 
the Carnot cycle is the simple 
rectangle ABOI) (Fig. 38), hav- 
ing the isothermals AB and CD 
at the temperatures T^ and T^ of 
the source and refrigerator, respec- 
tively. This geometrical rep- 
resentation affords an intuitive 
insight into the property of maxi- 
mum efficiency. Between the 
same isothermals let us assume 
some other form of cycle, as the 
trapezoidal cycle EBCD. For the ^ 
rectangular cycle the efficiency is 

heat transformed into work _ 
heat supplied 

For the trapezoidal cycle, likewise, the efficiency is 

area BEBC 



A, 


E, 




Fig. 38. 


area 


ABCD 


area 


A^ABB^ 



But 



DEBC 



area A^DEBB^ 
ABOD-AEB 



< 



ABCD 



A^BEBB^ A^ABB^ - AED A^AB^B 

that is, the efficiency of the trapezoidal cycle is less than that 
of the rectangular cycle. In the same way it can be shown 



136 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix 

"^ 
that any cycle lying wholly within the rectangular cycle ABCD 

has a smaller efficiency than the rectangular cycle. 

With a given source and refrigerator, the conditions of maxi- 
mum efficiency, which may be approached but never actually 
attained, are the following : 

1. The medium must receive heat from the source at the 
temperature of the source. No heat must be received at lower 
temperature. 

2. The medium must reject heat to the refrigerator at the 
temperature of the refrigerator. 

3. Provided the medium, source, and refrigerator are the 
only bodies involved in the transfer of heat, it follows from 1 
and 2 that the intermediate processes must be adiabatic, as any 
departure from the adiabatic would mean passage of heat to 

or from some body at a tem- 

A/r-, 7iB perature different from either 

the source or refrigerator. 

89. Isoadiabatic Cycles. — Let 

a cycle be formed with the iso- 
thermals AB and OB as in the 
Carnot cycle, but with the 
adiabatics replaced by similar 
curves BO and AB (Fig. 39) ; 
that is, curve BO is simply 
curve BA shifted horizontally 
a distance AB. Then AB = 
BO, as in the Carnot cycle. If 

the cycle is traversed in the clockwise sense, the heat entering 

the medium is 

Qda -^ Qab = area B^BAA^ + area A^ABB^, 

while the heat rejected by the medium is • 

Qbc + Qcd = area B^B 00^ + area O^ OBB^. 

The heat transformed into work is the same as in the Carnot 
cycle, for the area of the figure ABOB is equal to that of the 
Carnot rectangle. Now if the heat Q^^ represented by area 




Fig. 39. 



ART. 90] CLASSIFICATION OF AIR ENGINES 137 

D^DAA^ is taken from the source of heat, the efficiency of the 
cycle is 

_ heat transformed _ area ABOD 
heat taken from source area D^DABB^ ' 

and this is manifestly smaller than the efficiency of the Carnot 
cycle. Let it be observed, however, that 

Qhc = Qdai 

that is, area B^B CC^ = area D^BAA^. 

If the heat rejected by the medium during the process BC 
could be stored instead of thrown away, then this heat might 
be used again during the process DA, thus saving the source 
the heat Q^^. In this case we should have the following series 
of steps : 

1. Medium absorbs heat Q^ from source. 

2. Medium rejects heat Qj,^, which is stored. 

3. Medium rejects heat Q^^ to refrigerator. 

4. Medium absorbs the heat Q^^ (= Qbc} stored during 
step 2. 

Since in this case the source furnishes only the heat Qab-, the 
efficiency is 

area ABCB 
V = 



area A^ABB^' 

which is the same as that of the Carnot cycle. A cycle in 
which the adiabatics of the Carnot cycle are replaced by similar 
curves, along which the interchanges of heat are balanced, is 
called an isoadiabatic cycle. Any such cycle has the same ideal 
efficiency as the Carnot cycle. 

90. Classification of Air Engines. — Heat motors that employ 
air or some other practically perfect gas as a working fluid may 
be divided into two chief classes : (1) Motors in which the fur- 
nace is exterior to the working cylinder, so that the medium is 
heated by conduction through metal walls. (2) Motors in which 
the medium is heated directly in the working cylinder by the 
combustion of some gaseous or liquid fuel. These are called 
internal combustion motors. 

We may make a second division based on the manner in 



138 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix 

which the working fluid is used. In the closed-cycle type of 
motor, the same mass of air is used over and over again, fresh 
air being supplied merely to replace leakage losses. In the 
open-cycle type a fresh charge of air is drawn in at each stroke, 
and after passing through its cycle is discharged again into the 
atmosphere. 

Air motors of the first class, namely, those with the furnace 
exterior to the working cylinder, are usually designated as hot- 
air engines. Motors of this class are no longer constructed 
except in small sizes for pumping and domestic purposes ; they 
are, however, of historical interest, and besides they furnish iur 
structive illustrations of the application of the regenerative 
principle. We shall, therefore, describe briefly the two leading 
types of hot-air engines and give an analysis of the cycles. 



91. Stirling's Engine. — The engine designed by Robert 
Stirling in 1816, and bearing his name, is of the external fur- 
nace closed-cycle type. 
The general features of 
the engine are shown in 
Fig. 40. A displacer 
piston Q works in a cyl- 
inder Q, Between Q and 
an outer cylinder D is 
placed a regenerator RR^ 
made of thin metal plates 
or wire gauze. At the 
upper end of the cylinder 
is a refrigerator ilf, com- 
posed of a pipe coil through 
which cold water is made 
to circulate. At the lower 
end is the fire F. The 
piston Q is filled with some 
non-conducting material. The working cylinder A has free 
communication with the displacer cylinder. In the actual 
engine there are two displacer cylinders, one for each end of 
the working cylinder, which is double acting. 




Fig. 40. 



ART. 92] ERICSSON'S AIR ENGINE 139 

The action of tlie engine is as follows : Assume the working 
piston P to be at the beginning of its upward stroke and the 
displacer piston at the bottom of its cylinder. The air is, 
therefore, all in the upper part of the cylinder in contact with 
the refrigerator, and its state may be represented by the point 
D (Fig. 39). Now let the displacer piston be moved suddenly to 
the upper end of its cylinder. The air is forced through R 
and the perforations in into the lower end of the cylinder. 
The air remains at constant volume, since the piston P has not 
yet moved, and has received heat in passing through R. Hence 
the change of state is a heating at constant volume represented 
by DA in the diagram. The air now receives heat from the 
furnace and expands at constant temperature during the up- 
ward working stroke of piston P. This process is represented 
by AB. When the piston P reaches the upper end of its 
stroke, the displacer piston Q is suddenly moved to the bottom 
of the cylinder, thus forcing the air back through R into the 
refrigerator M. This again is a constant volume change and is 
represented by BQ. Lastly, during the return stroke the air is 
compressed isothermally, as represented by (7i), and heat is re- 
jected to the refrigerator. 

The ideal cycle is seen to be an isoadiabatic cycle with 
the adiabatics of the Carnot cycle replaced by constant-volume 
curves. The cycle given by the actual engine deviates consid- 
erably from the ideal cycle on account of the large clearance 
necessary between the two cylinders. 

A double acting Stirling engine of 50 i. h. p. was used for 
some years at the Dundee foundry, but was eventually aban- 
doned because of the failure of the regenerators. This 
engine had an efficiency of 0.3 and consumed 1.7 lb. of coal 
per i. h. p. 

92. Ericsson's Air Engine. — The Swedish engineer Ericsson 
made several attempts to design hot-air engines of considerable 
power. His large engines proved failures, however, because of 
their enormous bulk and the rapid deterioration of the regener- 
ators. The engines for the 2200-ton vessel Ericsson had four 
single-acting working cylinders 14 ft. in diameter and 6 ft. 



140 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix 

stroke and ran at 9 r.p.m. They developed 300 h.p. with a 
fuel consumption of 1.87 lb. of coal per h.p. -hour. 

The working of the Ericsson engine was substantially as fol- 
lows : A pump compressed air at atmospheric temperature into 
a receiver, whence it passed through the regenerator into a 
working cylinder. The pump was water-jacketed so as to act 
as a refrigerator. During the passage through the regenerator 
the air was heated at constant pressure. After the air was cut 
off in the working cylinder, it expanded isothermally, the nec- 
essary heat being furnished 
by a furnace external to the 
working cylinder. On the 
return stroke the air was dis- 
charged through the regener- 
ator at constant pressure. 

The p F^diagram is shown 
in Fig. 41. The pump cycle 
is DCFE, the motor cycle 
pj^ ^j ' EABF. The operations are 

as follows: 
(1) Compression in pump from C to i>, heat abstracted by 
pump water-jacket. (2) Discharge from pump to regenerator, 
represented by DE, (3) Suction of air into working cylin- 
der represented by EA. (4) Isothermal expansion from A to 
B^ during which air receives heat from furnace. (5) Dis- 
charge of air, represented by BF, (6) Suction of air into pump, 
represented by FQ. 

Deducting the work of the pump from that of the motor, the 
effective work is given by the diagram ABCD composed of the 
two isothermals and two constant-pressure lines. 

93. Analysis of Cycles. — The ideal cycles of the Stirling and 
Ericsson engines are isoadiabatic cycles. In the Stirling cycle 
the constant-volume lines DA and BO (Fig. 39) replace the 
adiabatics of the Carnot cycle. Using the ^>S'-plane we have 

Q^ = Ap^V^ log, 5 = ABT^M log, -p 

'^ a 'a 




ART. 941 HKATING BY INTERNAL COMBUSTION 141 

a, = Ap, F, log, ^' = - AMBT^ log, ^■ 



= AMB '" 



r^iog.-p-r.iog,^ 



But since T'„ = V^ and T' = V,,, 

AW= AMB (T,~T^) log, ^. 

Tlie heat Q^^ is taken from a regenerator, and therefore the 
heat Q^ alone is supplied from the source ; hence the efficiency- 
is 

For the Ericsson cycle DA and BO are constant-pressure 
lines and the analysis is essentially the same except that c^ is 
replaced by Cp. 

94. Heating by Internal Combustion.* — While the hot-air 
engine with exterior furnace should apparently be an efficient 
heat motor, experience has proved the contrary. The difficulty 
lies in the slow rate of absorption of heat by any gas. Even 
with high furnace temperatures and comparatively large heat- 
ing surfaces it has been found impossible to get a high tempera- 
ture in the working medium. Furthermore, if the air could be 
effectively heated, the metal surfaces separating the furnace from 
the hot medium would be destroyed; hence, while high tempera- 
ture of air is necessary for high efficiency, low temperature is 
necessary to secure the durability of the metal. 

These contradictory conditions are completely obviated by 
the method of heating by internal combustion. The rapid 
chemical action supported by the medium itself makes possible 
the rapid heating of large quantities of air to a very high 
temperature. The medium and the furnace being within the 
cylinder, the outer surface of the metal walls can be kept at 

* For an excellent discussion of this topic see Clerk's The Gas, Petrol, and 
Oil Engine, Revised Edition, Chapter I. 



142 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix 



low temperature by a water jacket, and consequently the inner 
surface can be exposed to the high temperature desired without 
danger of destruction. Furthermore, the low conductivity 
of gases becomes here an advantage as it prevents a rapid flow 
of heat from the medium to the cylinder walls. The low gas 
temperature of the hot-air engine results in a small effective 
pressure and makes the engine very bulky for the power 
obtained. The high temperature possible in the internal 
combustion motor, on the other hand, permits high effective 
pressures, and therefore gives a relatively small bulk per 
horsepower. 

95. The Otto Cycle. — The cycle of the well-known Otto 
gas engine has five operations as follows : 

1. The explosive mixture 
is drawn into the cylinder. 
Represented by EB, Fig. 42. 

2. The mixture is com- 
pressed, as represented by 
DA, 

3. The charge is ignited, 
causing a rise of temperature 
and pressure, as shown by AB. 

4. The gases in the cyl- 
inder expand adiabatically as 
shown by BC. 

5. The burned gases are expelled in part. Represented 
hy BE. 

In the case of the four-cycle Otto engine, each of the opera- 
tions 1, 2, 4, and 5 occupies one stroke of the piston, while 
operation 3 occurs At the beginning of a stroke. The cycle 
is completed in four strokes, whence the term four-cycle. 

It is customary in the analysis of gas-engine cycles to 
assume in the first instance that the medium is pure air 
throughout the cycle and that the air receives during the 
process AB an amount of heat equal to that developed by the 
combustion of the fuel in the actual cycle. This assumed ideal 
cycle is referred to as the air standard. 




Fig. 42. 



ART. 95] 



THE OTTO CYCLE 



143 



On the TO'-plane, the ideal cycle has the form shown in 
Fig. 43, AB and CD being constant volume curves. The 
medium in the state repre- 
sented by point A is heated at 
constant volume, as shown by 
the curve AB. For this pro- 
cess we have (assuming that e^ 
is constant) 

For the adiabatic expansion 
represented by BC, 




^'^~ k-1 



Ar 



X" 



-B 



Fig. 43. 



For the cooling at constant volume, represented by CD^ we 
have Q^ = Mc,{T^-T:)=-Mc,iiT,-T^-), 

Finally the medium is compressed adiabatically from B to A^ 
and for this change of state 



w. 



k-1 



The heat changed into work is 
The work of the cycle is 

ir= w^+ Tn„+ W^,+ W^= (F.n-F.T-e)-(y„F„-F.F.) 

k—\ 

It is easily shown that these results are identical. 
The efficiency is 

W Jc,\(iT,-T:)-iT,-T,y, 



(1) 

(2) 



t) = 



or 









(8) 



144 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix 



This expression for tj may be simplified as follows : From 
Fig. 43 we have 






hence, 
Therefore, 



^ 
T. 



or — ^ = ^ 



T. 



T.-T, 



Ta T,-T, 



or 



v=i- 



T 



C4) 



It appears, therefore, that the Otto cycle has the same efficiency 
as a Carnot cycle having the extreme temperatures T^, and T^ 
or the extreme temperatures T^ and T^ of the adiabatics, but a 
smaller efficiency than a Carnot cycle having 75, and T^ as 
extreme temperature limits. 

The expression for the ideal efficiency may be written in 
another convenient form. Since the curve DA represents an 
adiabatic process, we have 



whence 



,=1 



or 



v = l 



til 

pj 



(5) 



It appears from the last expression that the higher the com- 
pression pressure p^, the greater the ideal efficiency. 

If the ratio of volumes -^ be denoted by r, we have for the 
ideal efficiency the expression 



1 



(6) 



Example. If the air is compressed from 14.7 lb. to 45 lb., the ideal 



efficiency is 



1-(1M)" = 0.274. 



ART. 96] THE JOULE OR BRAYTON CYCLE 145 

If the compression is increased to 80 lb., the ideal efficiency is 



(Wf-- 



384. 





?i = o„(li-rj 


whence 




Since 


T;=r„ 


we have 





The temperature and pressure represented by the point B 
are readily calculated for this ideal case. Let q^ denote the 
heat absorbed per pound of air during the process AB ; then 



(7) 



(8) 

The value of q^ for a given fuel depends upon the heating 
value of the fuel and the weight of air required for the com- 
bustion of a unit weight of the fuel. 

96. The Joule or Brayton Cycle. — In the Otto type of motor, 
the fuel gas is mixed with air previous to compression, and 
when the mixture is ignited the combustion is so rapid as 
to produce an explosion; the heat is supplied, therefore, at 
practically constant volume. Another type of motor was first 
suggested by Joule and was developed in working form by 
Brayton (1872). In the Brayton engine the mixture of air 
and gas was compressed into a reservoir to a pressure of per- 
haps 60 lb. per square inch and from the reservoir flowed into 
the working cylinder, where it was ignited by a flame. A wire 
gauze diaphragm was used to prevent the flame from striking 
back into the reservoir. The mixture was thus burned quietly 
in the working cylinder during about one half the stroke of 
the piston, and by proper regulation of the admission valve the 
rate of combustion was so regulated as to give practically con- 
stant pressure during the period of admission. The ideal 
cycle of operations is as follows : 

1. Charge drawn into compressor cylinder, ED (Fig. 44). 

2. Adiabatic compression, DA. 



146 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix 



3. Expulsion at constant pressure from compressor, AF; 
simultaneous admission to motor cylinder, FB. The charge 

during the passage from 
A 7> compressor to motor is 

heated at constant pres- 
sure and the volume is 
thereby increased as in- 
dicated by AB. 

4. Adiabatic expansion, 
BO^ after cut off. 

5. Expulsion of burned 
gases, OF. 

The area FBAF repre- 
sents the negative work of the compressor, the area FBCE 
the work obtained from the motor ; hence, area ABCD repre- 
sents the net available work. 

On the T^-plane, the ideal Joule cycle has the same form as 
the Otto cycle (Fig. 43). The curves AB and CD, however, 
represent, respectively, heating and cooling at constant pressure. 
We have, therefore. 







2 J-c -l-d _ -[ _c _ 2^ 



Also, 






?i 



c,T,. 



+ 1. 



r. 



(1) 

(2) 
(3) 

(5) 



97. The Diesel Cycle. — The principle of gradual and quiet 
combustion as opposed to explosion was seized upon by Diesel 
in the design of the Diesel motor. In this motor air without 
fuel is compressed in the working cylinder to a pressure ap- 
proximating 500 lb. per square inch. The temperature at the end 
of compression is consequently higher than the ignition tempera- 
ture of the fuel. At the end of the compression stroke the 
fuel is injected into the air and at once burns. By proper 
regulation of the fuel supply, the air may be made to 



ART. 97] 



THE DIESEL CYCLE 



147 




Fig. 45. 



expand at practically constant 

pressure, or if desired, with 

falling pressure and nearly 

constant temperature. As in 

the Bray ton engine, govern- 
ing is effected by cutting off 

the fuel injection earlier or 

later. 

The ideal cycle of the Diesel 

engine is shown in Fig. 45. It 

resembles the Otto cycle except 

that the process AB in this case represents a constant pressure 

rather than a constant volume combustion. It was the original 

B aim of Diesel so to regulate the 

injection of fuel that a short 
period of combustion AM 
would be followed by isother- 
mal expansion MJV, the fuel 
being cut off at the point iV. 

On the IW-plane the ideal 
Diesel cycle is shown in 
Fig. 46, in which AB is ?i 
constant-pressure curve and 
CJ) a constant- volume curve. 
We have then 





M^- 


iV 


A 




G 


D 







o 



Fig. 46. 






J7 = 



(T,-Tj 



k\n-z 



(1) 

(2) 
(3) 

(4) 



If the cycle includes an isothermal process, as il/TV, we have 

Qam = Mc^{T^-T,\ (5) 



Q^, = AMBT^\oo-^, 



and 7) = 



m I ^%mn i Xc 
Vawi+ Vmn 



T.-T, 



hiT„-T:)^{h-\)T„\o^, 



(6) 
(7) 



148 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix 

98. Comparison of Cycles. — The three principal cycles are 
shown superimposed in Fig. 47. The minimum temperature 
at I) and maximum temperature at B are the same for all 

three. With this assumption 
it is seen that the Brayton 
cycle A' BCD has the largest 
area, the Otto cycle ABOD^ 
the smallest. Hence, between 
the same temperature limits 
and with the same maximum 
pressure pi,, the Brayton cycle 
is the most efficient, the Otto 
cycle the least efficient. Com- 
? paring the maximum volumes, 
it is seen that the Otto and 
Diesel cycles have the same 
maximum volumes F^, while the Brayton cycle requires a 
greater volume, as indicated by the point C. The Diesel 
cycle, therefore, combines the advantages of the high efficiency 
of the Brayton cycle due to the high compression pressure 
and the smaller cylinder volume of the Otto cycle. 




Fig. 47. 



99. Closer Analysis of the Otto Cycle. — In the preceding 
analysis of gas-engine cycles two assumptions have been made : 
(1) That the medium employed has throughout the cycle the 
properties of air. (2) That the specific heat of the medium is 
constant. While the approximate analyses based on these 
assumptions are of value in giving the essential characteristics 
of the various cycles, and an idea of their relative efficiencies, 
they give misleading notions regarding the absolute magnitudes 
of those efficiencies. To obtain the true value of the maximum 
possible efficiency of a gas-engine cycle, it is necessary to take 
account of the properties of the fuel mixture entering the cylin- 
der and of the mixture of the products of combustion after the 
fuel is burned. Making use of the principles and methods 
laid down in Chapter VIII, we may thus make an accurate 
analysis of any one of the cycles discussed in the preceding 
articles. The following example, the data for which are fur- 



ART. 100] AIR REFRIGERATION 149 

nislied by the example of Art. 85, shows such an analysis for 
the Otto cycle. 

Example. Determine the ideal efficiency of an Otto cycle in which the 
compression, combustion, and expansion follow the course described in the 
example of Art. 85. Compare this efficiency with the '^ air standard " 
efficiency under the same conditions. 

In the example quoted, the work of adiabatic compression was found to 
be 69,550 ft.-lb., the work of expansion 283,600 ft. -lb. These results refer 
to 1 lb. of the fuel mixture. The heating value of the fuel per pound was 
found to be 1632.2 B.t.u. ; hence the heating value per pound of fuel mix- 
ture is 1632.2 -^ 2.25 = 725.-4 B. t. u. The net work derived from the cycle 
per pound of mixture is 283,600 - 69,550 = 214,050 ft.-lb. Therefore, the 
efficiency is 

' J X 725.4 

The "air standard" efficiency depends upon the ratio of initial and final 

volumes, which ratio was found to be — = 0.1887. Hence, for this efficiency 

' 1 

we have 77 = 1 - 0.18870-4 = 0.487. 

The discrepancy between the two efficiencies is in a large measure due to 
the assumption of constant specific heat in the analysis of Art. 95. 

100. Air Refrigeration. — The term refrigeration is applied 
to the process of keeping a body permanently at a temperature 
lower than that of surrounding bodies. Since heat naturally 
flows from the surroundings to the body at lower temperature, 
this heat must be continually removed if the body is to remain 
permanently at its lower temperature. Hence a refrigerating 
machine has the office of removing heat from a body of low 
temperature and depositing it in some other convenient body 
of higher temperature. 

The operation of a refrigerating machine is thus precisely 
the reverse of tiie operation of the direct-heat motor; and if 
the cycle of a heat motor be traversed in the reverse direc- 
tion, it will give a possible cycle for a refrigerating machine. 
When air is used as a medium for refrigeration, the reversed 
Joule cycle is employed. Fig. 48 shows diagrammatically the 
arrangement of the refrigerating machine. Fig. 49 the ideal 
jt? F^diagram, and Fig. 50 the TO'-diagram . Air in the state A 
in the cold room is drawn into the compressor c and is com- 



150 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix 




Fig. 48. 



pressed adiabatically as indicated by AB. It then passes into 
the cooling coils, about which cold water circulates, and is 
cooled at constant pressure, as indicated by BO. In the state 

the air passes 
into the expansion 
cylinder e and is 
permitted to ex- 
pand adiabatically 
down to the pres- 
sure in the cold 
room, i.e. atmos- 
pheric pressure. 
The final state is 
represented by 
point D. Finally the air absorbs heat from the cold room, and 
its temperature rises to the original value T^- Referring to 
Fig. 49, the actual compression diagram is ABFE., while the 
diagram FODE taken clockwise is the diagram of the expan- 
sion cylinder. The net work done on the air is, therefore, 
given by the diagram ABCB. 

The Allen dense-air machine has a closed cycle and the air 
is always under a pressure much higher than that of the atmos- 
phere. Thus the pressure BA (Fig. 49) is perhaps 40 to 60, 
and the upper pressure, say 
200 lb. per square inch. The 
air, after expanding to the 
lower pressure, is led through 
coils immersed in brine and 
absorbs heat from the brine. 

In the following analysis of 
the air-refrigerating machine 
we shall assume ideal condi- 
tions. In the actual machine 
these conditions are to some 

extent modified. The compression and expansion are not truly 
adiabatic, and there is a drop in pressure between the cylinders 
due to frictional resistances in the coils. 

Let Q denote the heat absorbed from the cold body per 




Fig. 49, 



ART. 100] 



AIR REFRIGERATION 



151 



minute, and M the weight of air 
circulated per minute. Then 
since in passing through the cold 
body the temperature of the air is 
raised from T^ to T,, (Fig. 50), we 
have 

Q=^Me,iT,-T,-). (1) 

Let jOj denote the suction pres- 
sure of the compressor cycle 
(atmospheric pressure, in the 
case of the open cycle) and p^ 
the pressure at the end of com- 
prefesion ; then, assuming adiabatic compression, we have 

and if the pressure at cut-off in the expansion cylinder is also 
^2 (as in the ideal case), we have also 




By 



Fig. 50. 



(2) 



T. \pj 



whence 



Ic 

T, 






The work required per minute is 

area ABCD 
W= JQ X 



JQ 



n-y. 



(3) 
(4) 

(5) 



area C^DAB^ 

and the heat rejected to the cooling water, represented by the 
area B^BCC^ (Fig. 50), is 



Q + 



W 



^ rp 

J- n 



(6) 



The compressor cylinder draws in per minute iJf pounds of air 
having the pressure f^ and temperature T^. Denoting by N 
the number of working strokes per minute and by F^ the volume 
displaced by the compressor piston, we have for the ideal case 



152 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix 



or 

Likewise, the volume V^ of the expansion cylinder is given by 
the relation 

Example. An air-refrigerating machine is to abstract 600 B.t. u. per 
rainnte from a cold chamber. The pressure in the cold room is 14.7 lb. per 
square inch, and the air is compressed adiabatically to 65 lb. per square inch 
absolute. The temperature in the cold room is 36^ F. and the air leaves the 
cooling coils at 80° F. The machine makes 120 working strokes per minute. 
Required the ideal horsepower required to drive the machine, and the volumes 
of the compression and expansion cylinders. 

The first step is the determination of the temperature T^ at the end of 
expansion. From the relation 






d \pi- 

^^^^^^ r, = 539.6 (i|^y~':= 352.9. ' 

From (1) we obtain for the weight of air that must be circulated per minute 

M = . ^ = - ^00 ^17.52 lb. 

Cp( Ta - To) 0.24(495.6 _ 352.9) 

The work required per minute is 

W = JQ^^i-:^-^ = 778 X 600 x 539.6 - 352.9 ^ 246,950 ft. lb. , 
Ta 352.9 

and the horsepower under these ideal conditions is therefore 

246950 ^ y g 
33000 

For the volume of the compressor cylinder, we have 

J. 172.9 X 53.34 x 495.6 . ^ .. 

Vc = = l.o CU. It., 

120 X 14.7 X 144 
and for the volume of the expansion cylinder 

V,= F,^==1.8 X §^z= 1.29 CU. ft. 
Ta 495.6 

101. Air Compression.^ — ^Air at a pressure greater than that 
of the atmosphere is used extensively in engineering operations, 



ART. 101] 



AIR COMPRESSION 



153 



especially in mining, tunneling, and metallurgical processes. 
The compression of air may be effected by rotary fans and 
blowers or by piston compressors. In the piston compressor, 
air at atmospheric pressure is drawn into a cylinder through in- 
let valves and is then compressed upon the return stroke of the 
piston. When the desired pressure is attained, the outlet valves 
are opened and the air is discharged into a receiver. The ideal 
indicator diagram of an air 
compressor has, therefore, the 
form shown in Fig. 51. The 
line DA represents the drawing 
in of the air ; the curve AB rep- 
resents the compression from 
tlie lower pressure p^ to the 
receiver pressure p^; and BO 
represents the expulsion of the 
air at the higher pressure. It 
should be noted that the curve 

AB represents a change of state, while lines DA and BO 
represent merely change of locality ; thus BO represents the 
passage of the air (in the same state} from the compressor 
cylinder to the receiver. 

Let F^ denote the volume denoted by point -A, and F^ the 
volume after compression ; then the work of compression (area 
A.ABB,) is nn-P^V. ^ 

n — 1 




Fig. 51. 



TFl; 



assuming that the compression curve follows the law jt?F" = const. 
The w^ork of expulsion (represented by area B^BOO) is evidently 

Wb, = -p^V^, 

and the work done by the air during the intake (area ODAA^^ is 

Hence, the total work represented by the area of the diagram 



-^ (f 1^1 -ft ^2)- 



(1) 



154 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix 



From the relation p-^ ¥{" = p^, ^2"' ^^^ have 



Fo 



-'-■©■ 



(2) 



whence combining (1) and (2) we get 



w-l 



W 



n 



i^>^{^-(S)"]' , "-'' 



a formula that does not contain the final volume V^. 

For the temperature at the 
end of compression we have the 
usual formula 

n-\ 




T, 



Px 



(4) 



The action of the air com- 
pressor may be studied advanta- 
geously by means of the TS- 
diagram. Let the point A (Fig. 
52) represent the state of the 
air at the beginning of com- 
pression, and suppose that AB 
represents the compression pro- 
cess. Through B a line repre- 
senting the constant pressure 
^2 is drawn, intersecting at F an isothermal through A. It 
can be shown that the area A^ABFF^ represents the work W 
given by (1). Denoting by T^ the final temperature corre- 
sponding to point B^ we have 



Fig. 52. 



area A^ABB-^ = Me 



n 



5|(T. 



area B^BFF^ = Mc^, (2\ - T^), 



&TesiA^ABFF^=M(cp 



n 



k 



T{). 



(T.-T,-) 



= M- 



n 



n—1 



c^ — c 



71 

1 n 
Jn-1 



^ — ^^ (^MBT^ - MBT^) 



1 B 



iPiV.-p^v^y 



(5) 



ART. 102] WATER-JACKETING . 155 

Comparing (5) with (1), it is seen that the area under the 
curves AB and BD represents the heat equivalent of the 
work W. 

102. Water-jacketing. — Unless some provision is made for 
withdrawing heat during the compression, the temperature will 
rise according to the adiabatic law. Ordinarily the energy 
stored in the air due to its increase of temperature, that is, the 

energy Z/^ - C^i = Mo, (T^-T,), 

is never utilized because during the transmission of the air 
through the mains heat is lost by radiation and the temperature 
falls to the initial value. Hence 
a rise in the temperature during 
compression indicates a useless 
expenditure of work. The water 
jacket prevents in some degree 
this rise in temperature and 
decreases the work required for 
compression. The curve AU 
(Fig. 53) represents adiabatic o' ;; ^ 

compression. If the compres- 
sion could be made isothermal, the curve would be AF, less 
steep than AF^ and the work of the engine would be reduced 
per stroke by the area AEF, The water jacket gives the curve 
AB lying between AE and AF^ and the shaded area represents 
the saving in work. Because of the water jacket the value of 
the exponent n in the equation jt? F" = const, lies somewhere 
between 1 and 1.40. Under usual working conditions, n is 
about 1.3. 

Yon: any value of n the relation between the heat abstracted, 
work done, and change of energy is given by the proportion 

JQ:(U^- U^) : W= (k - n} : (1 - n) : (k - 1). 

This applies only to the compression AB not to the expulsion 
of the air represented by B O. 

The influence of the water jacket is shown more clearly by 
the TW-diagram, Fig. 52. The vertical line AF indicates adia- 
batic compression from p^ to jpg^ the horizontal line AF, isother- 




156 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap.ix 

mal compression, and the intermediate curve AB, compression 
according to the law pV^= const., with n between 1 and 1.4. 
The area A^ABB^ represents the heat abstracted from the air 
during compression, and the area ABB represents the work 
saved by the use of the jacket. A more efficient jacket would 
give a compression curve with its extremity lying nearer the 
point B. In the case of the isothermal compression represented 
by AF, the area A^AFF^ represents the heat absorbed from the 
air and also the work done on the air. These must necessarily 
be equivalent, since there is no change in the internal energy. 




Fig. 54. 



103. Compound Compression. — The excess of work required 
by the increase of temperature during compression may be obvi- 
ated in some measure by 
dividing the compression 
into two or more stages. 
Air is compressed from 
the initial pressure p^ to 
an intermediate pressure 
jt?', it is then passed 
through a cooler where 
the temperature (and con- 
sequently the volume) is 
reduced, and finally it is 
compressed from p' to the desired pressure p^^ In Fig. 54, 
DA represents the entrance of air into the cylinder, and AG, 
which lies between the adiabatic AB and the isothermal AF, 
the compression in the first cylinder. From Gr to ^the air is 
cooled at constant pressure in the intercooler. The curve HL 
shows the compression in the second cylinder, and the line 
LO the expulsion into the receiver. In a single cylinder the 
diagram would be ABCB; hence compounding saves the work 
indicated by the area BGHL. 

The saving is shown even more clearly if we use the TS- 
plane (Fig. 55). During the first compression AG the heat 
represented by the area A^AGG-^ is absorbed by the water 
jacket. Then the heat G^GHH^ is abstracted by the inter- 
cooler. During the second compression the heat H^HLL^ is 



ART. 103] 



COMPOUND COMPRESSION 



157 



abstracted by the water jacket, 
and finally the heat L^LFF^ 
is radiated from the receiver 
and main. As shown in the 
preceding article, the area 
A^AGrHLFF^ gives the work 
of the compressor. Evidently 
area BGrHL represents the 
work saved by compounding. 
If we take (3) of Art. 101, 
we find for the work done in 
the first cylinder 



T 


B 




^i ! 1 1 1 


O 


^\ Ly -Hi Gi ^x ' 



Fig. 55. 



n-l 



w,= 



n 



XPi'^y 



:)■} 



and for the work done in the second cylinder 



^2= T 

2 n—1 



p'V 



where V is the volume indicated by point H (Fig. 54). 
But since point J^is on the isothermal AF^ we have 

and, therefore, 



TK = 



n 



Vi 



n-l 
n 



The total work is, consequently. 



Tf,+ TFi 



PxV, 



2 — 



(1) 



p'J Ui. 

The work required is numerically a minimum when the 
expression 



(?) ■ -&;) 



has a maximum value. Note that p^ and p^ are fixed, while p^ 



158 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix 

is variable. Using the ordinary method of the calculus, we 
find that this expression is a maximum when 

P' = ^PiPt (2) 

Equation (2) is useful in proportioning the cylinders of a com- 
pound compressor. 

Referring to Fig. 55, we have 

With the condition expressed by (2) we have 

n -1 n—1 






n-1 
"2' 



and likewise, 

Hence, ^i = T^ ; 

that is, for a minimum work of compression the points Cr and L 
should lie on the same temperature level. The same statement 
applies to three-stage compression. 

104. Compressed-air Engines. — Compressed air may be used 
as a working fluid in a motor in substantially the same way 
p as steam. In fact, compressed air 
^^ B has in some instances been used 

in ordinary steam engines. The 
indicator diagram for the motor 
should approach the form shown 
in Fig. 5Q. With clearance and 
•^' ^' ^ compression, A'U' will replace 

o' '^ AJE. The work per stroke is 

Fig 56 

readily calculated in either case. 
The expansion curve BO may be taken as an adiabatic. 

105. TS-diagram of Combined Compressor and Engine. — The 

^>S^-diagram shows clearly the losses in a compressed-air system 
and the effects of various expedients employed to reduce such 



A' 




B 




\ 
\ 
\ 




\ 




\ 






\^ 


\ 

\ 






\C 




^, 








E' 




D 



ART. 105] COMBINED COMPRESSOR AND ENGINE 



159 




losses. In the following discussion we shall take up first an 
ideal case and afterwards several modifications that may be 
made. 

In Fig. 57, m represents the compressor diagram, n the 
motor diagram, both without clearance. Air in the state repre- 
sented by point A is 
taken into the com- 
pressor at atmos- 
pheric pressure and 
temperature. The 
compression, as- 
sumed here to be 
adiabatic, is repre- 
sented on the TS- 

plane by the vertical line AB (Fig. 58). The expulsion of 
the air into the receiver and thence into the main is merely a 
change of locality and does not itself involve any change of 
state ; hence, it is not represented on the ^>iS'-plane. However, 
the passage of the air along the main is usually accompanied 
by a cooling, and this is represented by ^O' (Fig. 58), the final 
point O representing the state of the air at the beginning of 
expansion in the motor. The adiabatic expansion to atmos- 
pheric pressure in the motor is 
represented by CD. This is 
accompanied by a drop in tem- 
perature which is given by the 
equation 







Fig. 58. 



The air discharged from the motor 
in the state I) is now heated at 
the constant pressure of the atmos- 
phere until it regains its original temperature T^- This heating 
is represented by DA. 

The complete process is a cycle of four distinct operations, 
two of which are adiabatic and two at constant pressure ; that 
is, the cycle is a reversed Joule cycle. The question now arises: 



160 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix 



what does the area ABCD of the cycle represent — something 
useful or something wasteful ? To answer this question let us 
recur to the original energy equation 

JQ= U^- U^+ W, 

and apply it to the air which passes through the cycle process 
just described. We have 

Work done on air = area of diagram m — — Wm- 
Work done by air = area of diagram n = -{- W^. 
Total work = W^ - TT^. 
Heat absorbed by air = area under DA. 
Heat rejected by air = area under BO. 
Total heat put into system = — area ABCD. 
Change of energy = Z7„ — 1/^=0. 
Hence, 



Jx 'dre^ABOD = W^ 



tf: 




that is, the area ABOD represents the difference between the 
work done by the compressor and the work delivered by the 

motor. Consequently it 
represents a waste, which 
is to be avoided as far as 
possible. 

Various modifications 
of the simple cycle of 
Fig. 58 are shown in 
Fig. 59. The effect of 
using a water jacket is 
shown at (a). The 
shaded area represents 
the saving. 

Figure 59 (5) shows 
the effect of reheating 
the air before it enters the motor. In the main the air cools, 
as indicated by BC^ but in passing through the reheater it is 
heated again at constant pressure, and the state point retraces 
its path, say to D. Then follows adiabatic expansion DE, and 
constant-pressure heating EA. This reheating saves work 





Fig. 59. 



ART. 105] COMBINED COMPRESSOR AND ENGINE 161 

represented by the area CDEF. It would be possible to carry 
I) to the right of B^ in which case the waste would become 
zero or even negative. The area CDEF does not, however, 
represent clear gain, as account must be taken of the heat 
expended in the process OD. 

In Fig. 59 ((?) is shown the effect of compound compression, 
and in Fig. 59 (c?) the effect of compound compression with 
a compound motor. In each case the shaded area represents 
the saving. 

It would not be difficult to represent also the loss of pressure 
in the main due to friction. 



EXERCISES 

1. Find the efficiency of a Stirling hot-air engine working under ideal 
conditions between the .temperatures 1340" F. and 140"" F. Find the weight 
of air that must be circulated per minute per horsepower. 

2. An air compressor with 18 in. by 24 in. cylinder makes 140 working- 
strokes per minute and compresses the air to a pressure of 52 lb. per square 
inch, gauge. Assuming that there is no clearance, find the net horsepower 
required to drive the compressor. Take the equation of the compression 
curve as p V^-^ = const. 

3. If 200 cu. ft. of air at 14.7 lb. is compressed to a pressure of 90 lb. per 
square inch, gauge, find the saving in the work of compression and expulsion 
by the use of a water jacket that reduces the exponent n from 1.4 to 1.27. 

4. Find the efficiency of the ideal Otto cycle (air standard) when the 
compression is carried to 120 lb. per square inch absolute. 

5. Draw a curve showing the relation between the efficiency of the Otto 
cycle and the compression pressure. Take values of p from 40 to 200 lb, 
per square inch. 

6. An air-refrigerating machine takes air from the cold chamber at a 
pressure of 40 lb. per square inch and a temperature of 20° F., and com- 
presses it adiabatically to a pressure of 200 lb. per square inch. The air 
is then cooled at this pressure to 80° F. and expanded adiabatically to 
40 lb. per square inch, whence it passes into the coils in the cold chamber. 
The machine is required to abstract 45,000 B. t. u. per hour from the cold 
room, (a) Find the net horsepower required to drive the machine, (h) If 
the machine makes 80 working strokes per minute, find the necessary 
cylinder volumes. 

7. Air is to be compressed from 14.7 lb. per square inch to 300 lb. per 
square inch absolute. If a compound compressor is used, find the interme- 
diate pressure that should be chosen. 



162 TECHNICAL APPLICATIONS. GASEOUS MEDIA [chap, ix 

8. In Ex. 7, the compression in each cylinder follows the law pV^-^ = 
const. Find the saving in work effected by compounding, expressed in per 
cent of the work required of a single cylinder. 

9. Using the results of Ex. 10-15 of Chapter VIII, find the efficiencies of 
the Otto cycle with the natural gas and the blast furnace gas, respectively, 
under the conditions stated. Compare these efficiencies with corresponding 
air standard efficiencies. 

10. On the T^-plane draw accurately an ideal Diesel cycle from the fol- 
lowing data: Adiabatic compression of air from 14.7 to 500 lb. per square 
inch absolute ; heating at constant pressure to a temperature of 2200° F. ; 
adiabatic expansion to initial volume ; cooling at constant volume to initial 
state. Calculate the ideal efficiency of the cycle. 

11. Modify the Diesel cycle of the preceding example by stopping the 
constant-pressure heating at 1600° F. and continuing with an isothermal 
expansion (as shown by MN, Fig. 46). Calculate the efficiency of this 
modified cycle. 

12. The ideal Lenoir cycle has three operations, as follows : heating of air 
at constant volume, adiabatic expansion to initial pressure (atmospheric), and 
cooling at constant pressure. Show the cycle on pV- and T^S-planes, and 
derive an expression for its efficiency. 

13. Let the expansion in the Otto cycle be continued to atmospheric 
pressure. Show the resulting cycle on pV- and TS-planes and derive an 
expression for the efficiency. 

REFERENCES 

Hot-air Engines 

Innes : Applied Thermodynamics for Engineers, 129. 
Zeuner : Technical Thermodynamics (Klein) 1, 340. 
Rankine : The Steam Engine (1897), 370. 
Ewing : The Steam Engine, 402. 

Gas-engine Cycles 

Clerk : Gas, Petrol, and Oil Engines, 67. 

Carpenter and Diederichs : Internal Combustion Engines, 65. 

Levin : The Modern Gas Engine, 43. 

Berry : The Temperature Entropy Diagram, 167. 

Innes: Applied Thermodynamics, 154. 

Peabody : Thermodynamics of the Steam Engine, 5th ed., 304. 

Lorenz : Technische Warmelehre, 421. 

Weyrauch : Grundriss der Warme-Theorie, 277. 



ART. 105] REFERENCES 163 

Air Refrigeration 

Innes: Applied Thermodynamics, 396. 

Ewing : The Mechanical Production of Cold, 38. 

Peabody : Thermodynamics of the Steam Engine, 5th ed., 397. 

Zeuner: Technical Thermodynamics, 384. 



Air Compression 



Peabody : Thermodynamics, 5th ed., 358. 
Innes : Applied Thermodynamics, 96. 



CHAPTER X 

SATURATED VAPORS 

106. The Process of Vaporization. — The term vaporization 
may refer either (1) to the slow and quiet formation of vapor 
at the free surface of a liquid or (2) to the formation of vapor 
by ebullition. In the latter case, heat being applied to the 
liquid, the temperature rises until at a definite point vapor 
bubbles begin to form on the walls of the containing vessel and 
within the liquid itself. These rise to the liquid surface, and 
breaking, discharge the vapor contained in them. The liquid, 
meanwhile, is in a state of violent agitation. If this process 
takes place in an inclosed space — as a cylinder fitted with a 
movable piston — so arranged that the pressure may be kept 
constant while the inclosed volume may change, the following 
phenomena are observed: 

1. With a given constant pressure, the temperature remains 
constant during the process ; and the greater the assumed pres- 
sure, the higher the temperature of vaporization. The tempera- 
ture here referred to is that of the vapor above the liquid. As a 
matter of fact, the temperature of the liquid itself is slightly 
greater than that of the vapor, but the difference is small and 
negligible. 

2. At a given pressure a unit weight of vapor assumes a 
definite volume, that is, the vapor has a definite density; 
and if the pressure is changed, the density of the vapor changes 
correspondingly^ The density (or the specific volume) of a 
vapor is, therefore, a function of the pressure. 

3. If the process of vaporization is continued at constant 
pressure until all the liquid has been changed to vapor, then if 
heat be still added to the vapor alone, the temperature will rise 
and the specific volume will increase ; that is, the density will 
decrease. 

164 



ART. 106] 



THE PROCESS OF VAPORIZATION 



165 



So long as any liquid is present the vapor has a constant maxi- 
mum density and a constant temperature. The vapor in this 
case is said to be saturated and the constant temperature corre- 
sponding to the pressure at which the process is carried on is 
the saturation temperature. If no liquid is present, and through 
absorption of heat the temperature of the vapor rises above the 
saturation temperature, the vapor is said to be superheated. 
The difference between the temperature of the vapor and the 
saturation temperature is called the degree of superheat. 

The process just described may be represented graphically 
on the jt?T^-plane. See Fig. 60. Consider a unit weight of 
liquid subjected to a pressure jt? represented by the ordinate of 
the line A' A" ; and let the 
volume of the liquid (de- 
noted by v'} be represented 
by A'. As vaporization 
proceeds at this constant 
pressure, the volume of 
the mixture of liquid and 
vapor increases, and the 
point representing the 
state of the mixture moves 
along the line A' A" . The 
point A" represents the 
volume v" of the saturated 
vapor at the completion 
segment A' A" represents 
Any point between A' and 
of a mixture of liquid and 
point depends on the ratio of the weight of the vapor to 
the weight of the mixture. Denoting this ratio by x^ we have 

A'M 

appears 




of 
the 



vaporization 
increase of 



therefore, 
volume v'^ 



the 



X = 



AA"' 



whence it 



A'\ as M^ represents the state 
vapor, and the position of the 



that at A', a; = 0, while at J.", 



x = l. This ratio x is often called the quality of mixture. 

If the mixture is subjected to higher pressure during vapor- 
ization, the state-point will move along some other line, as B'B". 
The specific volume indicated by jB'' is smaller than that indicated 
by A", The curve v", giving the specific volumes of the satu- 



166 



SATURATED VAPORS 



[chap. X 



rated vapor for different pressures, is called the saturation curve ; 
while the curve v\ giving the corresponding liquid volume, is 
the liquid curve. These curves v\ v" are in a sense boundary 
curves. Between them lies the region of liquid and vapor 
mixtures, and to the right of v" is the region of superheated 
vapor. Any point in this latter region, as E^ represents a state 
of the superheated vapor. 

107. Functional Relations. Characteristic Surfaces. — For a 
mixture of liquid and saturated vapor, the functional relations 
connecting the coordinates p^ v, and t are essentially different 
from the relation for a permanent gas. As explained in the 
preceding article, the temperature of the mixture depends 
upon the pressure only, and we cannot, as in the case of a 
gas, give p and t any values we choose. The volume of a unit 
weight of the mixture depends (1) upon the specific volume 
of the vapor for the given pressure and (2) upon the quality 
X. Hence we have for a mixture the following functional 
relations : 

t = fip-),oTp = F(t), (1) 

v^^ip,x). (2) 

With superheated steam, as with gases, p and t may be 
varied independently, and consequently the functional relation 
between p^ v, and t has the general form 

f(p, V, = 0. (3) 

The characteris^tic surface of a 
saturated vapor is shown in Fig. 61. 
It is a cylindrical surface jS whose 
generating elements cut the p^-plane 
in the curve p = F(t). These ele- 
ments are limited by the two space 
curves v' and v" ^ which when pro- 
jected on the jov-plane give the 
curves v\ v" of Fig. 60. The space 
curve v" is the intersection of the 
surface S and the surface for the 
Fig. 61. superheated vapor. 




ART. 108] PRESSURE AND TEMPERATURE 167 

108. Relation between Pressure and Temperature. — The rela- 
tion p = F{t^ between the pressure p and temperature ^ of a 
saturated vapor must be determined by experiment. To Reg- 
nault are due the experimental data for a large number of 
vapors. Further experiments on vi^ater vapor have been made 
by Ramsey and Young, by Battelli, and very recently by Hol- 
born and Henning. These last-mentioned experiments were 
made with the greatest accuracy and with all the refinements 
of modern apparatus ; they may, therefore, be regarded as 
furnishing the most reliable data at present available on the 
pressure and temperature of saturated water vapor. Experi- 
ments on other saturated vapors of technical importance, carbon 
dioxide, sulphur dioxide, ammonia, etc., have been made by 
Amagat, Pictet, Cailletet, Dieterici, and others. It is likely, 
however, that further experiments must be made before the 
data for these vapors are as reliable as those for water vapor. 

If the experimentally determined values of p and t be plotted, 
they will give the curve whose equation is p = f{t) (Fig. 61). 
To express this relation many formulas have been proposed, 
some purely empirical, some having a more or less rational 
basis. A few of these formulas are the following : 

1. Biofs Formula. — As used by Regnault, Biot's equation 
has the form 

log ^ = a — 5a" + ^/3% (1) 

where n = t— c. 

This formula is purely empirical. Having five constants, the curve 
may be made to pass through five experimentally determined 
points ; hence, the formula may be made to fit the experimental 
values very closely throughout a considerable range. The follow- 
ing are the values of the constants as given by Prof. Peabody : 

For Steam from 32° to 21.2° F., p For Steam from 212° to 428° F., p 

IN Pounds per Sqcare Inch. in Pounds per Square Inch. 

a = 3.125906 a = 3.743976 

log h = 0.611740 log h = 0.412002 

log c = 8.13204 - 10 log c = 7.74168 - 10 

log a = 9.998181 - 10 log a = 9.998562 - 10 

log /3 =0.0038134 log yS= 0.0042454 

n = f-S2 n = t-2U 



168 SATURATED VAPORS [chap, x 

2. Rankine^s Formula. — Rankine proposed an equation of 

the form 7^ p 

logp=A + f+±, (2) 

in which T denotes the absolute temperature. This formula 
has been much used in calculating steam tables, especially in 
England. Having but three constants, it is not as accurate 
as the Biot formula. The following are the values for the 
constants, when p is taken in pounds per square inch, and 
1^ = ^ + 460: 

^ = 6.1007; 5 = -2719.8; (7=400125. 

3. The DuprS-Hertz formula has the form 

\ogp = a-h\og T- ^. (3) 

This equation has been derived rationally by Gibbs, Bertrand, 
and others, and gives, with a proper choice of constants, results 
that agree well with experiment. Using the results of Reg- 
nault's experiments, Bertrand found the following values of the 
constant for various vapors (metric units). 

a h c 

Water 17.44324 3.8682 2795.0 

Ether 13.42311 1.9787 1729.97 

Alcohol ....... 21.44687 4.2248 2734.8 

Chloroform 19.29793 3.9158 2179.1 

Sulphur dioxide .... 16.99036 3.2198 1604.8 

Ammonia 13.37156 1.8726 1449.8 

Carbon dioxide .... 6.41443 - 0.4186 819.77 

Sulphur 19.1074 3.4048 4684.5 

4. BertrancCs Formulas. — Bertrand has suggested two equa- 
tions, namely : ^^ 

P=^-^ ^ (4) 

and p^k (-^)"- (5) 

The latter may be written in the more convenient form 

T 

log p = \ogk-n log rp _^ ' (^) 



ART. 108] 



PRESSURE AND TEMPERATURE 



169 



Bertrand's second formula (6) has the advantage over the 
others suggested of lending itself to quick and easy computa- 
tion. Furthermore, although it has but three constants, it 
gives results that agree remarkably well with the experiments 
of Holborn and Henning on water vapor. The constants are 
as follows (English units) : 

T= ^ + 459.6 

n = 50. 



Fkom 



From 90° - 237° 



From 238° - 420° F. 



5 = 140.1 5 = 141.43 5 = 140.8 

log^= 6.23167 log^= 6.30217 log k = 6.27756 

The agreement between observed and calculated values is 
shown in the following table. The maximum difference is 
one tenth of one per cent. . 



Temperature, C. 


PeessTtre in Mm. of Mercury 


Bertrand's Formula 


Experiments of 
Holborn and Henning 





4.577 


4.579 


10 


9.208 


9.205 


20 


17.511 


17.51 


30 


31.682 


31.71 


40 


55.121 


55.13 


50 


92.325 


92.30 


60 


149.21 


149.19 


70 


233.55 


233.53 


80 


354.97 


355.1 


90 


525.64 


525.8 


100 


760 


760 


110 


1075.2 


1074.5 


120 


1489.7 


1488.9 


130 


2025.2 


2025.6 


140 


2708.3 


2709.5 


150 


3566.7 


3568.7 


160 


4631.1 


4633 


170 


5935.2 


5937 


180 


7515 


7514 


190 


9409.1 


9404 


200 


11658 


11647 



170 SATURATED VAPORS [chap, x 

5. Marks' Equation. — Professor Marks has deduced an 
equation that gives with remarkable accuracy the relation 
between jp and 2^ throughout the range 32° F. to 706.1° F., the 
latter temperature being the critical temperature, as established 
by the recent experiments of Holborn and Banmann. The 
form of the equation is 

log p=. a- ~-cT+eT\ (7) 

The constants have the following values: a = 10.515354, b = 
4873.71, c = 0.00405096, e = 0.000001392964. 

109. Expression for — — In the Clapej^ron-Clausius formula 
for the specific volume of a saturated vapor, the derivative -^ 

CtJ/ 

is required. An expression for this derivative is obtained by 
differentiating any one of the equations (1) to (7) of Art. 108. 
Thus from (6), 

^ = np(-^ ly ""^P ; (1) 



whence 



log ^ = log nh + log JO - log 7^ _ log (2^ - h). 
at 



d'l 



Values of -^ are readily calculated since the terms log T, 

log {T — 5), and log p appear in the calculation of p from (6). 

110. Energy Equation applied to the Vaporization Process. — 

It is customary in estimating the energy, entropy, heat content, 
etc., of a saturated vapor to assume liquid at 32° F. (0°C.) as a 
datum from which to start. Thus the energy of a pound of 
steam is assumed to be the energy above that of a pound of 
water at 32° F. 

Suppose that a pound of liquid at 32° is heated until its 
temperature reaches the boiling point corresponding to the 
pressure to which the liquid is subjected. The heat required 
is given by the equation 

q' = gc^dt, (1) 

where c' denotes the specific heat of the liquid. This process 



ART. 110] 



VAPORIZATION PROCESS 



171 




M 




is represented on the 2W-plane by a curve AA' (Fig. 62). 
The ordinate OA represents the initial absolute temperature 
32 + 459.6 = 491.6, the ordinate A^A' the temperature of va- 
porization given by the relation t=f(^p}^ and the area OAA'A^ 
the heat q' absorbed by the liquid. This heat q' is called the 
heat of the liquid.* 

When the temperature of vaporization is reached, the liquid 
begins to change to vapor, the temperature remaining constant 
during the process. A definite quantity of heat, dependent 
upon the pressure, is required to change the liquid completely 
into vapor. This is called the 
heat of vaporization and is de- 
noted by the symbol r. In Fig. 
62, the passage of the state- 
point from A' to A'^ represents 
the vaporization, and the heat 
r is represented by the area 
A^A'A"A^. For a higher pres- 
sure the curve AB' represents 
the heating of the liquid and 
the line B' B" the vaporization. 

During the heating of the 
liquid the change in volume is 
very small and may be neg- 
lected ; hence, the external work done is negligible also, and 
substantially all of the heat q' goes to increase the energy of 
the liquid. During the vaporization, how^ever, the volume 
changes from v' (volume of 1 lb. of liquid) to v" (volume of 
1 lb. of saturated vapor). Since the pressure remains constant, 
the external work that must be done to provide for the increase 
of volume is L=p(iv"-v'). (2) 

According to the energy equation, the heat r added during 
vaporization is used in increasing the energy of the system and 
in doing external work. Hence, the difference 

r- AL = r-Ap (v" - v'') (3) 

* In this chapter symbols with primes, c', q', v', s', etc., are used for the liquid ; 
symbols with double primes, c'', q", v" , s", etc., for the saturated vapor. 



A,B, 



Fig. 62. 



B2A2E1 



172 SATURATED VAPORS [chap, x 

is the heat required to increase the energy of the unit weight 
of substance when it changes from liquid to vapor. This heat 
is denoted by p and is called the internal latent heat. Since 
during the vaporization the temperature is constant, there is no 
change of kinetic energy ; it follows that p is expended in in- 
creasing the potential energy of the system. The heat equiva- 
lent of the external work, namely, Ap (y^' — v'}, is called the 
external latent heat, and for convenience may be denoted by i/r. 

We have then , , .^ 

r^p-h'f. (4) 

The total heat of the saturated vapor is evidently the sum of 
the heat of the liquid and the heat of vaporization. Thus, 

q" =q' -hr, 
or q" = qf -^ p -[- yjr. (5) 

Comparing (5) with the general energy equation, it is evident 

that the sum q' -\- p gives the increase of energy of the saturated 

vapor over the energy of the liquid at 32° F. Denoting this 

by u", we have ... , , ,^. 

-^ Au" = q' +/0- (o) 

If the vaporization is not completed, the result is a mixture 
of saturated vapor and liquid of quality x ix= J, as indi- 

cated by the point Jf (Fig. 60 and 62). In this case the heat 
required to vaporize the part x is xr heat units and the total 
heat of the mixture, which may be denoted by q^^ is given by 

q^=q' -{- xr 

= q' + xp + xyjr, (7) 

The energy of the mixture (per unit weight) above the energy 
of water at 32° F. is, therefore, given by the relation 

Au^ = q' -{- xp, (8) 

and the external work done is 

L, = Jxylr. (9) 

If heat is added at constant pressure, after the vaporization is 
completed, the vapor will be superheated. The state-point will 
move along the curve A"^ (Fig. 48), and the heat Cp(t^—t") 



ART. 112] THERMAL PROPERTIES OF WATER VAPOR 173 

represented by the area A^A" EE^ will be added. Here c^ de- 
notes the mean specific heat of the superheated vapor, t^ the 
final temperature, and t" the saturation temperature correspond- 
ing to the pressure p. The total heat corresponding to the 
point ^and represented by the area OAA' A" EE^ is, therefore, 

qe=-q' + r-{-e^(t,-t"). (10) 

If Vg denotes the final volume, and u^ the energy above liquid 
at 32° F., then the external work for the entire process is 

X = ^(^;,-^;'), (11) 

and, therefore, 

Au,=^q,-Ap(:v,-v'). (12) 

111. Heat Content of a Saturated Vapor. — By definition we 
have for the heat content of a unit weight of saturated vapor 

i^' = A(u" +pv"^ = q' ^ p + Apv^\ (1) 

Since the total heat is 

q" = q' + p + Ap(y"-v'), (2) 

it appears that i" is larger than q'' by the value of the term 
Apv' . As v\ the specific volume of water, is small compared 
with v'\ the term Apv' may be neglected except for very high 
pressures, and q" and i" may be considered equal. 

In most of the older steam tables values of q" were given ; 
in the more recent tables, the values of i" instead of q" are 
usually tabulated. 

112. Thermal Properties of Water Vapor. — From the relation 

q" =q' ^-r, 

it appears that if any two of the three magnitudes q'\ q\ r are de- 
termined by experiment, the third may be found by a combina- 
tion of those two. Various experiments have been made to 
determine each of these magnitudes for the range of temperature 
ordinarily employed, and as a result several empirical formulas 
have been deduced. Naturally the greatest amount of attention 
has been given to water vapor, and we may consider the proper- 
ties of this medium as quite accurately known at the present 
time. Ammonia, sulphur dioxide, and other vapors have not 



174 



SATURATED VAPORS 



[chap. X 



been studied with the same completeness, and their properties 
are as yet only imperfectly known. 

In the sections immediately following we shall give briefly the 
results of the latest and most accurate experiments on water 
vapor. 

113. Heat of the Liquid. — Denoting d the specific heat of 
water, the heat of the liquid above 32° F. is given by the re- 

l-«°" 5' = £.'di. ' (1) 

If the specific heat c?' were constant at all temperatures, this 
equation would reduce to the simple form g' = e'(t — 32). As 
a matter of fact, however, c' is not constant, and its variation 
with the temperature must be known before (1) can be used to 
calculate q'. Between 0° C. and 100° C. (32°-212° F.) the 
experiments of Dr. Barnes may be regarded as the most trust- 
worthy. Taking c' = 1 at a temperature of. 17.5° C, the fol- 
lowing values are given by Griffiths as representing the results 
obtained by Barnes. 



Temperature 




Temperature 






Specific Heat 




Specific Heat 


C. 


F. 




C. 


F. 







32 


1.0083 


55 


131 


0.9981 


5 


41 


1.0054 


60 


140 


0.9987 


10 


50 


1.0027 


65 


149 


0.9993 


15 


59 


1.0007 


70 


158 


1.0000 


20 


68 


0.9992 


75 


167 


1.0007 


25 


77 


0.9978 


80 


176 


1.0015 


30 


86 


0.9975 


85 


185 


1.0023 


35 


95 


0.9974 


90 


194 


1.0031 


40 


104 


0.9973 


95 


203 


1.0040 


45 


113 


0.9974 


100 


212 


1.0051 


50 


122 


0.9977 









These values are shown graphically in Fig. 63. From them 
values of q' may be obtained by means of relation (1). 

In the actual calculation of the tabular values of q\ the fol- 
lowing method may be used advantageously. Since the specific 
heat c' does not differ greatly from 1, let 

c'=^ 1 + k, 



ART. 114] 



LATENT HEAT OF VAPORIZATION 



175 



-^here ^ is a small correction term. Then for c^ we have 

If now values of k are plotted as ordinates with correspond- 
ing temperatures as abscissas, the values of the integral \ kdt 
may easily be determined by graphical integration. 

For temperatures above 212° F. the only available experi- 
ments giving the heat of 
the liquid are those of 
Regnault and Dieterici. 
The results of these ex- 
periments are somewhat 
discordant and unsatis- 
factory. Fortunately, 
we have for the range 
212° to 400° F. reliable 
formulas for the total 
heat 5-" and the latent 
heat r, and we may therefore determine 5' from the relation 



1.008 



1.006 



1.004 



.1.002 



1.000 



0.998 



1 




r : 




\ 








. r 


/ 


5 


y 


, v 


/" 


5 : 


/_ 


, ^ 


/ 


' 0° \§0° 40° 


60°/ 80° lio 


\_ 


^^""^ 


^^ _--■' 













Fig. 63. 



?' = 



r. 



114. Latent Heat of Vaporization. — The latent heat of water 
vapor for the range 0° to 180° C. (32°-356° F.) has been accu- 
rately determined by direct experiment. The results of the 
experiments of Dieterici at 0° C, Griffiths at 80° and 40° C, 
Smith over the range 14°-40° C, and Henning over the range 
30°-180° C. show a remarkable agreement, all of the values 
lying on, or very near, a smooth curve. The observed values 
are given in the third column of the following table. As the 
thermal units employed by the different investigators were not 
precisely the same, all values have been reduced to a common 
unit, the joule. 

It is readily found that a second-degree equation satis- 
factorily represents the relation between r and t. Taking r in 
joules, the following equation gives the values in the fourth 
column of the table : 



r= 2265.6 - 2.7405(f - 100) - 0.003389(^ - 100)2. (1) 



176 



SATURATED VAPORS [chap, x 

LATENT HEAT OF WATER, IN JOULES 





Tempera- 
ture, C. 


Latent Heat 


Difference 
Pee (Jent 


Observed 


Calculated 


Dieterici 





2493.8 


2495.8 


-0.08 


Griffiths 


30.00 
40.15 


2429.3 
2403.6 


2430.8 
2407.5 


-0.06 
-0.16 


Smith 


13.95 
21.17 

28.06 
39.80 


2467.6 
2451.2 
2435.0 

2405.8 


2466.3 
2450.5 
2435.2 

2408.3 


+ 0.05 
+ 0.03 
- 0.01 
-0.10 


Henning, 

First Series .... 


30.12 
49.14 
64.85 
77.34 
89.29 
100.59 


2424.8 
2385.3 
2343.0 
2313.7 
2285.6 
2254.2 


2430.6 
2386.2 
2347.7 
2316.0 
2284.6 
2254.0 


-0.24 
-0.04 
-0.20 
-0.10 
+ 0.05 
+ 0.01 


Henning, 

Second Series . . . 


102.34 

120.78 
140.97 
160.56 
180.72 


2248.7 
2200.2 
2134.2 
2077.0 
2018.6 


2249.2 
2197.2 
2137.6 
2077.2 
2012.3 


-0.02 
+ 0.14 
-0.16 
-0.01 
+ 0.31 



The differences between the observed values and those calcu- 
lated from this formula are shown in the last column. 

The mean calorie is equivalent to 4.184 joules ; hence, divid- 
ing the constants of Eq. (1) by 4.184, the resulting equation 
gives r in calories. This equation is readily changed to give 
r in B. t. u. with t in degrees F. We thus obtain finally 

r = 970.4 - 0.655 (t - 212) - 0.00045 (t - 212)2. (2) 

This formula may be accepted as giving quite accurately the 
latent heat from 32° F. to perhaps 400° F.* 

* Henning has proposed an exponential formula for r. As modified by Dr. 
Davis, this formula becomes in English units 

r = 139(689 - ty-^^^, 
or log r := 2.14302 + 0.315 log (689 - 0- 

The exponential formula has the advantage of making the value of r = at the 



ART. 116] SPECIFIC VOLUME OF STEAM 177 

115. Total Heat. Heat Content. — For the temperature range 
32° to 212° F. the total heat q" is obtained from the relation 
q" z= q' -\-r. As has been shown, values of q' and of r can be 
accurately determined for this range. For temperatures be- 
tween 212° and 400°, we are indebted to Dr. H. N. Davis for 
the derivation of a formula for the heat content of saturated 
vapor of water. The earlier experiments of Regnault led to 
the formula f = 1091.7 + 0.305 (t - 32), 

which has been extensively used in the calculation of tabu- 
lar values. By making use of the throttling experiments of 
Grindley, Griessmann, and Peake, Dr. Davis* has shown that 
Regnault's linear equation is incorrect, and that a second-degree 
equation of the form 

q^' = a-{-h(t- 212) + c(t- 212)2 
may be adopted. Dr. Davis obtains for the heat content i" 
the formula 

f = 1150.4 + 0.3745(^-212) - 0.000550-212)2. (3) 

From this formula the total heat q^^ is readily determined from 
the relation q" = i^' — Apv' . It is found, however, that slight 
changes in the constants are desirable in view of Henning's sub- 
sequent experiments on latent heat. The modified formula 

i^' = 1150.4 + 0.35 - 212) - 0.000333 (t - 212)2 (4) 
may be accepted as giving with reasonable accuracy values of 
{" for the range 212° to 400° F. 

116. Specific Volume of Steam. — The specific volume v" of 
a saturated vapor at various pressures may be determined 
experimentally. For water vapor accurate measurements of 
v'' for temperatures between 100° and 180° C. have been made 
by Knoblauch, Linde, and Klebe. It is possible, however, to 
calculate the volume v" from the general equations of thermo- 
dynamics ; and the agreement between the calculated values 
and those determined by experiment serves as a valuable check 

critical temperature, 689° F. At the higher temperatures it doubtless gives more 
accurate values than the second-degree formula. See Proceedings of the Amer. 
Acad, of Arts and Sciences 45, 284. 

* Trans. Am. Soc. of Mech. Engs. 30, 1419, 1908. See Art. 164 for a dis- 
cussion of the method employed in the derivation of formula (3). 

N 



178 



SATURATED VAPORS 



[chap. X 



on the accuracy with which the factors entering into the theo- 
retical formula have been determined. 

The general equation (Art. 56) / 



\dtjv 



(1) 



applies to any reversible process. Let us apply it to the pro- 
cess of changing a liquid to saturated vapor at a given constant 
temperature. For a saturated vapor, the partial derivative 

-^] is simply the derivative -^, and this is a constant for any 
otjv at 

given temperature (Art. 107). Hence, for the process in ques- 
tion, we have (since dT= 0^ 



dp 



_ AT^ 



dt Jv dt 



v'), 



(2) 



(3) 



But in this case q is the heat of vaporization r ; hence we have 

" — ' _ ^ 1 _ Jr 1 
^ ^ ~ ATdp~ Tdp' 
dt dt 

This is the Clapeyron-Clausius formula for the increase of vol- 
ume during vaporization. 

Having for any temperature the derivative — ^ (Art. 109) 

and the latent heat r, the change of volume v" — v' is readily 
calculated. The following table shows a comparison between 
the values of v" determined experimentally by Knoblauch, 
Linde, and Klebe, and those calculated by Henning from the 
Clapeyron equation, using the values of r determined from his 
own experiments. The third line gives values of v" calculated 
from the characteristic equation of superheated steam. (See 
Art. 132.) 





Specific Volumes, Cu. Meters per Kg. 


100° 


120° 


140° 


160° 


180° C. 


Experimental .... 

Henning 

From the equation for 
superheated steam . . 


1.674 
1.673 

1.673 


0.8922 
0.8912 

0.8915 


0.5091 

0.5078 

0.5084 


0.3073 
0.3071 

0.3071 


0.1913 
0.1947 

0.1945 



ART. 117] ENTROPY OF LIQUID AND OF VAPOR 179 

The relation between the pressure and specific volume v" of 
saturated steam may be represented approximately by an equa- 
tion of the form /^ ^ q^ ^^^ 

Zeuner, from the values of v'^ given in the older steam tables, 
deduced the value m = 1.0646. Taking the more accurate 
values of v" given in the later steam tables, we find 

^=1.0681, (7=484.2. 

117. Entropy of Liquid and of Vapor. — During the process 
of heating the liquid from its initial temperature to the tem- 
perature of vaporization the entropy of the liquid increases. 
Thus, referring to Fig. 62, if the initial temperature be 32° F., 
denoted by point J., and if the temperature be raised to that 
denoted by J.', the increase of entropy of the liquid is repre- 
sented by Ovlp the heat of the liquid by area OAA'A^. 

Since dq' = c'dT, we have as a general expression for the 
entropy s' of the liquid corresponding to a temperature T, 

J 491.6 T J 491.6 T ' 

If the specific heat c' is given as a function of jP, the inte- 
gration is readily effected. In the case of water, where the 
specific heat varies somewhat irregularly, as shown by the 
table of Art. 115, the following expedient may be used. Put 
c' = 1 -{- k ; then ^ is a small correction term that is negative 
between 63° and 150° F. and positive elsewhere. From (1) we 
have, therefore. 

The first term is readily calculated and the small correction 
term may be found by graphical integration. This method was 
used in calculating the values of s' in table I. 

The increase of entropy during vaporization, represented by 

V 

A' A" (Fig. 62), is evidently the quotient ^- Hence the en- 
tropy of the saturated vapor in the state A" is 

i" = «' + f • (3) 



180 SATURATED VAPORS [chap, x 

For a mixture of quality x, as represented by the point M^ the 
entropy is 

« = «' + f- ^ (4) 

118. Steam Tables. — The various properties of saturated 
steam considered in the preceding articles are tabulated for 
the range of pressure and temperature used in ordinary tech- 
nical applications. Many such tabulations have appeared. 
The older tables based largely upon Regnault's data are now 
known to be inaccurate to a degree that renders them value- 
less. The recent tables of Marks and Davis* and of Peabody,f 
however, embody the latest and most accurate researches on 
saturated steam. 

Table I at the end of the book has been calculated from 
the formulas derived in Arts. 108-116. The values differ but 
little from those obtained by Marks and Davis. The first col- 
umn gives the pressures in inches of mercury up to atmospheric 
pressure, and in pounds per square inch above atmospheric 
pressure ; the second column contains the corresponding 
temperatures. Columns 3 and 4 give the heat content of the 
liquid and saturated vapor, respectively. The values in col- 
umn 3 may be taken also as the heat of the liquid q' ; similarly, 
column 4 may be considered as giving the total heat q'' of the 
saturated vapor. As we have seen, the difference between i" 
and q" is negligible except at high pressures. 

119. Properties of Saturated Ammonia. — Several tables of 
the properties of saturated vapor of ammonia have been pub- 
lished. Among these may be mentioned those of Wood, Pea- 
body, Zeuner, and Dieterici. The values given by the different 
tables are very discordant, as they are for the most part obtained 
by theoretical deductions based on meager experimental data. 
For temperatures above 32° F. the values obtained by Dieterici 
as the result of direct experiment are most worthy of confidence. 

Dieterici determined experimentally the specific volume v^' 
of the saturated vapor for the temperature range 0° to 40° C. 

* Marks and Davis, Steam Tables and Diagrams^ Longmans, 1908. 
t Peabody, Steam and Entropy Tables^ J. Wiley and Sons, 1908. 



ART. 119] PROPERTIES OF SATURATED AMMONIA 181 

(32° to 104° F.) and also for the same range the specific heat / 
of the liquid ammonia. The formula deduced by Dieterici for 
specific heat is, for the Fahrenheit scale, 

c' = 1.118 + 0.001156 (t - 32). (1) 

From this formula, the heat of the liquid q' and the entropy of 
the liquid s' are readily calculated by means of the relations 



q' = Ce'dt, s' = C<^'~ 



The relation between pressure and temperature is given by 
the experiments of Regnault. The results of these experiments 
are expressed quite accurately by Bertrand's formula 

log p = 5. 87S95 - 50 log ^Z^^U' ^^^ 

Above 32°, having Dieterici's experimental values of v'^ and 



d[ 
the Clapeyron-Clausius formula 



from (2) the derivative — ^, we may find the latent heat r from 



r = A (y'^ - y') T^ . (See Art. 116.) (3) 

For temperatures below 32° we have neither v'^ nor r given 
experimentally; hence for this region values of various prop- 
erties can only be determined by extrapolation, and the ac- 
curacy of the results thus obtained is by no means assured. In 
calculating the values of table III the following method was 
used. The values of r for temperatures above 32° were calcu- 
lated by means of (3). It was found that these values may be 
represented quite accurately by the equation 

log r = 1.7920 - OA log (266 - 0, (4) 

in which 266° is the critical temperature of ammonia. (See p. 
176, footnote.) Formula (4) was assumed to hold for the range 
32° to — 30° ; and from the values of r thus obtained values of 
v^' were calculated by means of the Clapeyron relation (3). 

120. Other Saturated Vapors. — Several saturated vapors in 
addition to the vapors of water and ammonia have important 
technical applications. Sulphur dioxide and carbon dioxide in 



182 SATURATED VAPORS [chap, x 

particular are used as media for refrigerating machines. The 
properties of the former fluid have been investigated by Cailletet 
and Mathias, those of the latter by Amagat and Mollier. The 
results of these investigations are embodied in tables.* 

The properties of several vapors of minor importance have 
also been tabulated, the data being furnished for the most part 
by Regnault. These include ether, chloroform, carbon bisul- 
phide, carbon tetrachloride, aceton, and vapor of alcohol. f 

121. Liquid and Saturation Curves. — If for various tem- 
peratures the corresponding values of s', the entropy of the 
liquid, be laid off as abscissse, the result is a curve s'. Fig. 62. 
This is called the liquid curve. If, likewise, values of 

T 

be laid off as abscissae, a second curve s" is obtained. This 
is called the saturation curve. 

As already stated (Art. 106), any point between the curves s' 
and s'' represents a mixture of liquid and vapor, the ratio x de- 
pending upon the position of the point. It is possible, there- 
fore, to draw between the curves s' and s" a series of constant-a; 
lines. Each of the horizontal segments A' A" ^ B'B'\ etc., is 
divided into a convenient number (say 10) of equal parts and 
corresponding points are joined by curves. The successive 
curves, therefore, are the loci of points for which x — 0.1, 
x= 0.2, etc. 

The form of the saturation curve has an important relation 
to the behavior of a saturated vapor. For nearly all vapors, 
the curve has the general form shown in Fig. 62 ; that is, the 
entropy s" decreases with rising temperature. In the case of 
ether vapor, however, the entropy increases with rising tem- 
perature and the curve has, therefore, the same general direc- 
tion as the liquid curve s'. 

122. Specific Heat of a Saturated Vapor. — Referring to the 
saturation curve of Fig. 62, suppose the state-point to move 

* For tables of the properties of saturated vapor of SO2 and COo in English 
units, see Zeuner's Technical Thermodynamics^ Klein's translation, Part IL 

t See Peabody's Steam and Entropy Tables^ or Zeuner's Technical Thermo- 
dynamics^ Part II. 



ART. 122] SPECIFIC HEAT OF A SATURATED VAPOR 183 

from A" to B" . This represents a rise of temperature of the 
saturated vapor during which the vapor remains in the satu- 
rated condition. The process must evidently be accompanied 
by the withdrawal of heat represented by the area A^A'^ B"B^ ; 
and the reverse process, fall in temperature from B" to A!\ is 
accompanied by the addition of heat represented by the same 
area. It appears, therefore, that along the saturation curve 

the ratio — ^ is negative (except in the case of ether) ; that is, 

the specific heat of a saturated vapor is, in general, negative. 

An expression for the specific heat c" of the saturated vapor 
may be obtained as follows. The entropy of the saturated 
vapor is given by the equation 

b" = »'+^; (1) 

hence the change of entropy corresponding to a change of 
temperature is obtained by differentiating (1), thus 



<^- 



(2) 



But _ rf«' = ^. (3) 



and similarly for the saturation curve, 

, „ e"dT 
ds = — — -. 
T 

Substituting these values ds' and ds" in (2), the result is 



(4) 











•"-'■+ '■/j.' 


IW' 




or 








''''*jt- 


r 
T- 




But 


since 


c' = 


dq 
~ dT' 


(5) may be written 
d(q' + r) 


r 






" - dT 


T 




or 








dT r 






whei 


re q" -- 


-/ 


-h r is tlie total heat of the saturated 


vapor, 



(5) 



(6) 



184 SATURATED VAPORS [chap, x 

The derivative -^y- is readily found when an expression for 

q^' is known. Thus for water vapor above 212°, we have 
qf =a-\-b{t- 212) - e(t - 212)2 . 



whence 



g=5-2<^-212), 



where b = 0.S5 and c = 0.000333. 

At 212°, we have, for example, 

c" = 0.35 - - = 0.35 ^^IM__= _ 1.095. 

T 212 + 459.6 

123. General Equation for Vapor Mixtures. — Let heat be 
added to a unit weight of mixture of liquid and saturated 
vapor, of which the part x is vapor and the part 1 — a; is 
liquid. In general, the temperature T and quality x will 
change ; hence the heat added is the sum of two quantities : 
(1) the heat required to increase the temperature with x 
remaining constant ; (2) the heat required to increase x with 
the temperature constant. The first is evidently c'0. — x)dT 
+ c"xdT ; and the second is rdx ; hence we have 

dq=cX^-x)dT+c"xdT-^rdx (1) 

as the general differential equation for the heat added to a 
mixture. 

From (1) the general expression for the change of entropy 
of a mixture is given by 

^. = I = lO^-^^:L^dT + ldx. (2) 

The fact that ds is an exact differential leads at once to the 
relation ^ r- ... . „ ^ n / \ 

= U^\> (8) 



dx 



V(l — a;) 4- e"x 
T 



whence 



dT\T 
T dT\T/ 



or c" = c'-^ — --. (4) 

the relation that was obtained in Art. 122. 



ART.124] VARIATION OF X DURING ADIABATIC CHANGES 185 




'If 



124. Variation of x during Adiabatic Changes. — Let the point 
A" (Fig. 64) represent the state of saturated vapor as regards 
pressure and temperature. Adiabatic expansion will then be 
represented by a vertical line A" E^ the final point E being at 
lower temperature. Adiabatic compression will be shown by a 
vertical line A" G-. With a saturation curve of the form 
shown, it appears that during adiabatic expansion some of the 
vapor condenses, while adiabatic compression results in super- 
heating. If the state-point is originally at M so that x is some- 
what less than 1 (say 0.7 or 0.8), 
then adiabatic expansion is ac- 
companied by a decrease in x, 
adiabatic compression by an in- 
crease of X. 

If the saturation curve slopes 
in the other direction, as in the 
case of ether, the conditions just 
stated will, of course, be reversed. 

Adiabatic expansion of the 
liquid is represented by the line 
A'F ; evidently some of the 
liquid is vaporized during the 
process. If the mixture is originally mostly liquid, as indicated 
by a point iVnear the curve «', then adiabatic expansion results 
in an increase of x, adiabatic compression in a decrease of x. 

For a given pressure there is some value of x for which an 
indefinitely small adiabatic change produces no change in x ; 
in other words, at this point the constant-a? curve has a vertical 
tangent. For this point we have evidently dq^O and dx = 0, 
and the general equation (1), Art. 123, becomes 



Fig. 64. 



lc'(l-x)-\-c'x'\dT=(), 



whence 



or 



X = 



0, 


(1) 




(2) 


c' 
c' - c"' 


(3) 



Tlie locus of the points determined by (3) is a curve n (Fig. 64), 



186 SATURATED VAPORS [chap, x 

called the zero curve. Along this curve we have from the 
general equation 

dq = rdx ; (4) 

that is, all the heat entering the mixture is expended in vapor- 
izing the liquid. The zero curve is of little practical importance. 
The change of the quality x during the adiabatic expansion 
of a mixture is readily calculated by means of the entropy 
equation. In the initial state, the entropy of the mixture is 



x.r 



and in the final state it is 



1^ 






But for an adiabatic change s^ = s^; therefore, we have the 



x^r 



relation ^^^ + :^ =. .^^ + ^, (5) 

in which x^ is the only unknown quantity. 

125. Special Curves on the TS-plane. — The region between 
the liquid and saturation curves may be covered with series of 
curves in such a way that the position of the point represent- 
ing a mixture indicates at once the various properties of the 
mixture. 

In the first place, horizontal lines intercepted between the 
curves s' and s" are lines of constant temperature, also lines of 
constant pressure ; while vertical lines are lines of constant 
entropy. 

Lines of constant quality, Xy, x^^ 2^3, .. . may be drawn as 
explained in Art. 121. 

Curves of constant volume may be drawn as follows : The 
volume of a unit weight of mixture whose quality is x is given 
by the equation 

V = x(v^' — v') + v', (1) 

V — v^ 

whence a; =-77 7- C^^ 

v" — v' ^ ^ 

Suppose that the curve for some definite volume (say v = 5 cu. 
ft.) is to be located. For different pressures p^, p^^ p^, . . . 
the saturation volumes v^^', v^", Vg", . . . are known from the 



ART. 125] SPECIAL CURVES ON THE T/S-PLANE 



187 



tables. Substituting successively these values bi v'^ in (2), 
values of x^ as x^, x^^ x^^ . . . corresponding to the pressures 
Pv Pv Ps^ ' ' ' ^^^^ ^® found. The value of v' may be taken 
as constant for all pressures. The value of x^ locates a definite 
point on the p^ line, that of X2 a point on the p^ line, etc. The 
locus of these points is evidently a. curve, any point of which 
represents a mixture having the given volume v ; hence it is a 
constant- volume curve. 

In a similar manner curves of constant energy u may be 
located. Since u = q'+xp, (3) 



we have x = 

For given pressures pp p^^ . 



(*) 



X. = 



_u-q{ 



Xn 



etc. 



Pi P2 

Values of q' and p for different pressure are given in the table, 
and therefore for a given ^t, values of x^ x^^ , , . are readily 
calculated. These locate points on the corresponding j?-lines, 
and the locus of the points is 
the desired constant-'i* curve. 

By the same process may be 
drawn curves of constant total 
heat, 

q= q' -\- xr = const. 

or curves of constant heat 
content 

i — i' -{■ xr = const. 

In Fig. Q^^ the various curves 
are shown drawn through the ^ p^^ g^ 

same point P. From the general 

course of the curves the behavior of the mixture during a 
given change of state may be traced. Thus : (1) If a, mixture 
expands adiabatically, v increases but jo, T^ ^t, and i decrease. 
The quality x decreases as long as the state-point lies to the 
right of the zero curve. (2) If a mixture expands isody- 
namically (u— const.), v, s, and x increase, j9, T^ and i decrease. 




188 SATURATED VAPORS [chap, x 

(3) If heat is added to a mixture at constant volume, p^ T^ s, 
rr, w, and i all increase. 

Exercise. On cross-section paper draw liquid and saturation curves 
for water vapor, taking values of s' and s'l from the steam table. Then 
draw the curves v = 2, v = 10, y = 40 cu. ft. Also draw the curves w = 600 
B. t. u., u = 800 B. t. u. 

126. Special Changes of State. — Certain of the curves de- 
scribed in preceding articles represent important changes of 
state of the mixture of saturated vapor and liquid. The prin- 
cipal relations governing some of these changes will be de- 
veloped in this article. It is assumed that the system remains 
a mixture during the change, that is, that the path of the state- 
point is limited by the curves s' and s'L 

(a) Isothermal^ or Constant Pressure^ Change of State. — Let 
x-^ denote the initial quality, x^ the final quality. Then the 
initial volume is 

v^ = x^(v" — v'y + v' 

and the final volume is 

The change in volume is therefore 

^2 - ^1 = (^2 - ^i)(^"- v'), (1) 

and the external work is 

W=p(y^ - v^) = p(v" - v^)(x^ - x{). (2) 

The change of energy is 

u^-u^ = Jp(x^-x^}, (3) 

and the heat absorbed is 

q = r(x^-x^). (4) 

These equations refer to a unit weight of mixture. 

Example. At a pressure of 140 lb., absolute, the volume of one pound 
of a mixture of steam and water is increased by 0.8 cu. ft. The change of 

quality is ^'' ~ ^^ = — ==0.2514. The external work is 

^ -^ v"-v> 3.199-0.017 

140 X 144 X 0.8 = 16,128 ft.-lb. 

The increase of energy is Jp(x2 - x^) = 778 x 786.1 x 0.2514 = 153850 ft.-lb. ; 
and the heat absorbed is r (x^ — x{) = 869 x 0.2514 = 218.5 B. t. u. 



ART. 126] SPECIAL CHANGES OF STATE 189 

(5) Change of State at Constant Volume, — Since the volumes 
v^ and ^2 are equal, we have 

^ii< - ^') = ^M' - ^')^ (5) 

where v^' and v.^' are the saturation volumes corresponding to 
the pressures p^ and p^, respectively. From (5) the quality x^ 
in the final state may be determined. The external work TF'is 
zero ; hence we have for the heat absorbed 

q = A(u^ - u{) = (q^' + x^p^) - (^/ - x^p^) . (6) 

Example. A pound of a mixture of steam and water at 120 lb. pressure, 
quality 0.8, is cooled at constant volume to a pressure of 4 in. of mercury. 
Required the final quality and the heat taken from the mixture. 

From (5) 

^^ ^ x,(v," - V') ^ 0.8(3.724 - 0.017) ^ ^ ^^^^^ 
v^" - v' 176.6 

Therefore 

q = 311.9 + 0.8 X 795.8 - (93.4 -f 0.0167 x 959.5) = 839.2 B. t. u. 

(c) Adiahatic Change of State. For a reversible adiabatic 
change the entropy of the mixture remains constant ; hence we 
have 

from which equation the final quality x^^ can be found. Having 
x^^ the final volume v^ per unit weight is 

v = x^{v^'-v')-{-v'. (8) 

Since the heat added is zero, the external work is equal to the 
decrease in the intrinsic energy of the mixture. That is, 

W= u^-u^ = JlCq^' + x^p^) - {q^ + x^p^)^ . (9) 

Example. Three cubic feet of a mixture of steam and water, quality 
0.89, and having a pressure of 80 lb. per square inch, absolute, expands 
adiabatically to a pressure of 5 in. Hg. The final quality, final volume, 
and the external work are required. 

From the steam tables we find the following values : 





Q 


P 


s' 


T 


V" 


For /> = 80 lb. 


281.8 


819.6 


0.4533 


1.1667 


5.464 


For p = o in. Hg. 


101.7 


953.7 


0.1880 


1.7170 


143.2 



190 SATURATED VAPORS [chap, x 

The weight of the mixture is 

M = = ^ = 0.6167 lb. 

x^{v" - v') + v' 0.89(5.464 - 0.017) + 0.017 

From (7), the quality x^ in the second state is given by the relation 

0.4533 + 0.89 x 1.1667 ^ 0.1880 + 1.7170^:2, 

whence a;2 = 0.759. 

The volume in the second state, neglecting the insignificant volume of the 
liquid, is 

Fg = 0.6167 X 0.759 x 143.2 = 67.02 cu. ft. 

Finally, the external work is 

PF = 778 X 0.6167 [(281.8 + 0.89 x 819.6) - (101.7 + 0.759 x 953.7)] = 89,086 

ft.-lb. 

(c?) Isodynamic Change of State. If the energy of the mix- 
ture remains constant, we have 



or q^ + x^p^ = q^ + x^p^. (10) 

From (10) the final value of x is determined, and the final 
volume is then found from (8). 

For the isodynamic change, the heat added to the mixture is 
evidently equal to the external work. There is no simple way 
of finding the work. As an approximation, an exponential 

curve 

p^v^ = pv"^ (11) 

may be passed through the points jt?^, v^, and jOg^ ^'2' ^^^ ^^^ 
value of n can be found. This curve will approximate to the 
true isodynamic on the p?;-plane, and the external work will 
then be approximately 

W=P-Jh.::ilA. (12) 

n — 1 

In practice the isodynamic of vapor mixtures is of little 
importance. 

127. Approximate Equation for the Adiabatic of a Vapor Mix- 
ture. — In certain investigations, especially those relating to the 
flow of steam, it is convenient to represent the relation between 
p and V during an adiabatic change by an equation of the form 

pV-=a (1) 



ART. 127] APPROXIMATE EQUATION OF ADIABATIC 



191 



The value of the exponent n is not constant, but varies with the 
initial pressure, the initial quality, and also with the final 
pressure ; and at best the equation is an approximation. 
Rankine assumed for n the value J^ for all initial conditions. 
Zeuner, neglecting the influence of initial pressure, gave the 



formula 



n = 1.035 + 0.1 X. 



(2) 



Mr. E. H. Stone,* using the tables of Marks .and Davis, has 
derived the relation 

n = 1.059 - 0.000315 p + (0.0706 + 0.000376^>. (3) 

The following table gives values of n calculated from (3). 



Initial 




Initial Pressure in Labor per Square Inch, Absolute 


Quali- 
ty 






20 


40 


60 


80 


100 


120 


140 


160 


180 


200 


220 


240 


100 


1.131 


1.132 


1.133 


1.134 


1.136 


1.137 


1.138 


1.139 


1.141 


1.142 


1.143 


1.145 


0.95 


1127 


1.128 


1.128 


1.130 


1.131 


1.131 


1.132 


1.133 


1.134 


1.135 


1.136 


1.137 


0.90 


1.123 


1.123 


1.124 


1.124 


1.125 


1.125 


1.126 


1.126 


1.127 


1.127 


1.128 


1.129 


0.85 


1.119 


1.119 


1.119 


1.119 


1.120 


1.120 


1.120 


1.120 


1.120 


1.120 


1.120 


1.121 


0.80 


1.115 


1.115 


1.114 


1.114 


1.114 


1.114 


1.113 


1.113 


1.113 


1.113 


1.112 


1.112 


0.75 


1.111 


1.110 


1.110 


1.109 


1.109 


1.108 


1.107 


1.106 


1.106 


1.105 


1.104 


1.104 


0.70 


1.108 


1.106 


1.105 


1.104 


1.103 


1.102 


1.101 


1.100 


1.099 


1.098 


1.097 


1.096 


0.65 


1.104 


1.102 


1.101 


1.099 


1.098 


1.096 


1.095 


1.093 


1.092 


1.091 


1.089 


1.088 


0.60 


1.100 


1.098 


1.096 


1.094 


1.093 


1.091 


1.089 


1.087 


1.085 


1.083 


1.081 


1.080 


0.55 


1.096 


1.093 


1.092 


1.089 


1.087 


1.085 


1.083 


1.080 


1.078 


1.076 


1.074 


1.072 


0.50 


1.092 


1.089 


1.087 


1.084 


1.082 


1.079 


1.077 


1.074 


1.071 


1.069 


1.066 


1.064 



Having the initial values p-^^ F^, and a;^, and the final pressure 
jt?2i the final volume F^ is found approximately from (1), the 
appropriate value of n being taken from the table. The exter- 
nal work is found approximately by the usual formula for the 
change represented by (1), namely, 

^i^i-i^2^2 



W 



n — \ 



(4) 



Example. Taking the data of the example of Art. 126 (c), we have 
p^ = 80, Fj = o, x^ = 0.89, whence n = 1.123. The final pressure is 5 in. Hg. 
= 2.456 lb. per square inch. Hence from (1) 



80 \iAZi 



66.78 cu. ft.. 



.2.456/ 
* Graduating thesis, University of IlHnois, 1910. 



192 SATURATED VAPORS [chap, x 

and Tr = 144 x§^iil^^Mi2i^i56^ 88,974 ft.-lb. 

0.12o 

Comparing these results with the results obtained by the exact method, 
it appears that the volume Fg is about 0.36 per cent smaller and the work 
W about 0.13 per cent smaller. Hence the approximation is sufficiently 
close for all practical purposes. 

EXERCISES 

1. From Bertrand's equation calculate the pressure of steam corre- 
sponding to the following temperatures: 60°, 250°, 400° F. 

2. Find the values of the derivative -^ for the same temperatures. 

dt 

3. Using the results of Ex. 1 and 2, find the specific volumes for the 
given temperatures. 

4. Find (a) the latent heat, (b) the total heat of saturated steam, at a 
temperature of 324° F. 

5. Calculate the latent heat of steam, (a) by the quadratic formula (2), 
Art. 114 ; (b) by the exponential formula (see footnote, p. 176) for the tem- 
peratures 220° F. and 380° F. Compare the results. 

In the following examples take required values from the steam table, 
p. 315. 

6. Find the entropy, energy, heat content, and volume of 4.5 lb. of a 
mixture of steam and water at a pressure of 120 lb. per square inch, quality 
0.87. 

7. Find the quality and volume of the mixture after adiabatic expan- 
sion to a pressure of 16 lb. per square inch. 

8. Find the external work of the expansion. 

9. Using the data of the preceding examples, calculate the volume and 
work by means of the approximate exponential equation jo F" = C. 

10. A mixture, initial quality 0.97, expands adiabatically in a 12 in. by 
12 in. cylinder from a pressure of 100 lb. per square inch, gauge, to a pressure 
of 10 lb. per square inch, gauge. Find the point of cut-off. 

11. The volume of 6.3 lb. of mixture at a pressure of 140 lb. per square 
inch is 17.2 cu. ft. Find the quality of the mixture ; also the entropy 
and energy of the mixture. 

12. The mixture in Ex. 11 is cooled at constant volume to a pressure of 
20 lb. per square inch. Find the final value of x and the heat abstracted. 

13. At a pressure of 180 lb. per square inch the volume of 2 lb. of a 
mixture of steam and water is increased by 0.9 cu. ft. Find the increase of 
quality, increase of energy, heat added, and external work. 

14. A mixture of steam and water, quality 0.85, at a pressure of 18 lb. 
per square inch, is compressed adiabatically. Find the pressure at which 



ART. 127] EXERCISES 193 

the water is completely vaporized. Find also the work of compression 
per pound of mixture. 

15. Steam at a pressure of 80 lb. per square inch expands, remaining sat- 
urated until the pressure drops to 50 lb. per square inch. Find approxi- 
mately the heat that must be added to keep the steam in the saturated 
condition. 

16. Water at a temperature of 352*^ F. and under the corresponding 
pressure expands adiabatically until the pressure drops to 30 lb. per square 
inch. Find the per cent of water vaporized during the process. Find the 
work of expansion per pound of water. 

17. Two vessels, one containing M^ lb. of mixture at a pressure p^ and 
quality X]_, the other M, lb. at a pressure p^ and quality x^^ are placed in 
communication. No heat enters or leaves while the contents of the vessels 
are mixing. Derive equations by means of which the final pressure ps and 
final quality xg may be calculated. 

18. Let 1 lb. of mixture at a pressure of 20 lb. per square inch, quality 
0.96, enter a condenser which contains 20 lb. of mixture at a pressure of 3 in. 
Hg., quality 0.05. Assuming that no heat leaves the condenser during the 
process, find the pressure and quality after mixing. 

REFERENCES 
Pressure and Temperature of Saturated Vapors 

Regnault: Mem. de I'lnst. de France 21, 465. 1847. Rel. des exper. 2. 

Henning: Wied. Ann. (4) 22, 609. 1907. 

Holborn and Henning : Wied. Ann. (4) 25, 833. 1908. 

Holborn and Baumann : Wied. Ann. (4) 31, 945. 1910. 

Kisteen : The Locomotive 26, 85, 183, 246 ; 27, 54 ; 28, 88. 

These articles contain a very complete account of the experiments 

of Regnault, Holborn and Henning, and Thiesen. 
Chwolson : Lehrbuch der Physik 3, 730. 

Gives comprehensive discussion of the many formulas proposed for the 

relation between the pressure and temperature of various vapors. 
Preston : Theory of Heat, 330. 

Marks and Davis : Steam Tables and Diagrams, 93. 
Peabody : Steam and Entropy Tables, 8th ed., 8. 
Marks : Jour. Am. Soc. Mech. Engrs. 33, 563. 1911. 

Properties of Saturated Steam 
(a) Specific Heat of Water. Heat of Liquid 

Regnault : Mem. de I'Inst. de France 21, 729. 1847. 
Dieterici : Wied. Ann. (4) 16, 593. 1905. 
P>arnes : Phil. Trans. 199 A, 149. 1902. 
o 



194 SATURATED VAPORS [chap, x 

Rowland : Proc. Amer. Acad, of Arts and Sciences 15, 75 ; 16, 38. 1880- 

1881. 
Day: Phil. Mag. 46, 1. 1898. 
Griffiths: Thermal Measurement of Energy. 
Marks and Davis : Steam Tables and Diagrams, 88. 

(b) Latent Heat 

Regnault : Mem. de ITnst. de France 21, 635. 1847. 

Griffiths : Phil. Trans. 186 A, 261. 1895. 

Henning: Wied. Ann. (4) 21, 849, 1906; (4) 29, 441, 1909. 

Dieterici : Wied. Ann. (4) 16, 593. 1905. 

Smith : Phys. Rev. 25 145. 1907. 

(c) Total Heat 

Davis: Proc. Am. Soc. of Mech. Engrs. 30, 1419. 1908. Proc. Amer. 

Acad. 45, 265. 
Marks and Davis : Steam Tables and Diagrams, 98. 

(d) Specific Volume 

Fairbairn and Tate : Phil. Trans. (1860), 185. 

Knoblauch, Linde, and Klebe : Mitteil. iiber Forschungsarbeit. 21, 33. 1905 

Peabody : Proc. Am. Soc. Mech. Engrs. 31, 595. 1909. 

Peabody : Steam and Entropy Tables, 8th ed., 12. 

Marks and Davis : Steam Tables and Diagrams, 102. 

Davis : Proc. Am. Soc. Mech. Engrs. 30, 1429. 

Properties of Refrigrating Fluids 
(a) Ammonia 

Dieterici : Zeitschrift fiir Kalteindustrie. 1904. 
Jacobus: Trans- Am. Soc. Mech. Engrs. 12, 307. 
Wood: Trans. Am. Soc. Mech. Engrs. 10, 677. 
Peabody : Steam and Entropy Tables, 8th ed., 27. 
Zeuner: Technical Thermodynamics (Klein) 2, 252. 
Lorenz : Technische Warmelehre, 333. 

(b) Sulphur Dioxide 

Cailletet and Mathias : Comptes rendus 104, 1563. 1887. 
Lange : Zeitschrift fiir Kalteindustrie 1899, 82. 
Mathias : Comptes rendus 119, 404. 1894. 
Miller : Trans. Am. Soc. Mech. Engrs. 25, 176. 
Wood : Trans. Am. Soc. Mech. Engrs. 12, 137. 
Zeuner : Technical Thermodynamics 2, 256. 



ART. 127] REFERENCES ' 195 

(c) Carbon Dioxide 

Amagat : Comptes rendus 114, 1093. 1892. 
Mollier : Zeit. fur Kalteiiidustrie 1895, 66, 85. 
Zeauer: Technical Thermodynamics 2, 262. 

General Equations for Vapors. Changes of State 

Zeuner : Technical Thermodynamics 2, 53. 
Weyrauch : Grundriss der Warme-Theorie 2, 33. 
Preston : Theory of Heat, 650. 
Berry : Temperature Entropy Diagram, 43. 



CHAPTER XI 

SUPERHEATED VAPORS 

128. General Characteristics of Superheated Vapors. — The 

nature of a superheated vapor has been indicated in Art. 106, 
describing the process of vaporization. So long as a vapor is 
in immediate contact with the liquid from which it is formed it 
remains saturated, and its temperature is fixed by the pressure 
according to the relation ^ = /*(p). When vaporization is com- 
pleted, or when the saturated vapor is removed from contact 
with the liquid, further addition of heat at constant pressure 
results in a rise in temperature. If t^ denotes the saturation 
temperature given by t^ —f(^p^ and t the temperature after su- 
perheating, the difference ^ — ^^ is the degree of superheat. Thus 
for steam at a pressure of 120 lb. per square inch, tg= 311. 3° J5^; 
hence if at this pressure the steam has a temperature of 460°, 
the degree of superheat is 460° - 341.3° = 118.7°. 

As soon, therefore, as a vapor passes into the superheated 
state, the character of the relation between the coordinates jt?, v, 
and t changes. The temperature is freed from the rigid con- 
nection with the pressure that obtains in the saturated state, 
and p and t may be varied independently . The volume v of 
the superheated vapor depends upon both p and t thus taken as 
independent variables ; that is, 

2; = (/>(^, 0, (1) 

as in the case of a perfect gas. The form of the characteristic 
equation (1) for a superheated vapor is, however, less simple 
than that of the gas equation pv = BT. 

The state described by the term " superheated vapor " lies 
between two limiting states ; the saturated vapor on the one 
hand, and the perfect gas, obeying the laws of Boyle and Joule, 
on the other. The characteristic equation therefore should 

196 



ART. 129] 



CRITICAL STATES 



197 



be of such form as to reduce to the equation of the perfect 
gas, as the upper limit is approached and to give the proper 
values of jt?, v^ and t of saturated vapor when the lower limit 
is reached. In the case of compound substances like water 
or ammonia, however, one disturbing element is introduced 
at very high temperatures. The vapor may to some extent 
dissociate ; thus steam may in part split up into its components 
hydrogen and oxygen, ammonia into nitrogen and hydrogen. 
Nernst has found for example that at a pressure of one atmos- 
phere 3.4 per cent of water vapor is dissociated at a temperature 
of 2500° C. Manifestly the existence of dissociation must in- 
fluence the relation between the variables p^ v, and t. However, 
at the temperatures and pressures with which we are concerned 
in the technical applications of thermodynamics, the amount of 
dissociation is entirely negligible, and the characteristic equation 
may be assumed to hold for all temperatures within the range 
of ordinary practice. 



129. Critical States. — The region between the limit curves 
v'^ v'' (Fig. 60) or s\ s" (Fig. 62) is the region of mixtures of 
saturated vapor and liquid. 
The fact that these two curves 
approach each other as the tem- 
perature is increased suggests 
that a temperature may be 
reached above which it is im- 
possible for a mixture of liquid 
and vapor to exist. Let it be 
assumed that the two limit 
curves merge into each other 
at the point IT (Fig. 66^, and o" 
thus constitute a single curve, 
of which the liquid and saturation curves, as we have previously 
called them, are merely two branches. The significance of this 
assumption may be gathered from the following considerations. 

Let superheated vapor in the initial state represented by 
point A (Fig. 66 and 67) be compressed isothermally. Under 
usual conditions, the pressure will rise until it reaches the pres- 




FiG. (JG. 



198 



SUPERHEATED VAPORS 



[chap. XI 



sure of saturated vapor corresponding to the given constant 
temperature ^, and the state of the vapor will then be represented 
by point B on the saturation curve. Further compression at 
constant temperature results in condensation of the saturated 
vapor, as indicated by the line BC If the liquid be compressed 

iso thermally, the volume will be 
decreased slightly as the pres- 
sure rises, and the process will 
be represented by curve CD. 
The isothermal has therefore 
three distinct parts: along AB 
the fluid is superheated vapor, 
along BQ ^ mixture, and along 
OD a liquid. If the initial tem- 
perature be taken at a higher 
value t\ the result will be similar 
except that the segment B'C will 
be shorter. If the limit curves 
meet at point H^ it is evident that the temperature may be 
chosen so high that this horizontal segment of the isothermal 
disappears ; in other words, the isothermal lies entirely outside 
of the single limit curve. 

In Fig. QQ the segment BO represents the difference v^' — v' 
between the volume v^' of saturated vapor and the volume v' of 
the liquid; and in Fig. 67, the area B^BCO^ represents the la- 
tent heat r of vaporization. For the isothermal t^ that passes 
through IT^ the segment BO reduces to zero; hence, for this 
temperature and all higher temperatures, we have 




Fig. G^ 



V — V 



0, or v' 



and 



r = 0. 



V' = 



The second result also follows from the first when we consider 
the Clapeyron equation 

JrJ_ 

Tdp. 
dT 

The experiments of Andrews show that the condition just 
described may be actually attained. The isothermals for carbon 



ART. 129] 



CRITICAL STATES 



199 



dioxide as determined by Andrews are shown in Fig. 68. For 
t = 13.1° and 21.5° C. the horizontal segments corresponding 
to condensation are 
clearly marked. For 
^=31.1° the horizontal 
segment disappears and 
there is merely a point 
of inflexion in the 
curve. At 48.1° the 
point of inflexion dis- 
appeared, and the iso- 
thermal has the general 
form of the isothermal 
for a perfect gas. 

The temperature % 
was called by Andrews 
the critical tempera- 
ture. It has a definite 
value for any liquid. 
The pressure p^ and 
volume Vc indicated by the point H are called respectively the 
critical pressure and critical volume. Values of t^ and p^ fo^ 
various substances are given in the following table : 



p 

110 

Atm. 






















\ 












100 






\ 
















\ 


48.f 










90 




uV 




\ 














vg5.5° 


^ 


\ 








80 




v^ 


— ^o 


N 


\ 












81.1"N 


^ 


\ 


\ 






70 








\ 


^ 


N^ 


\ 












N 


^ 


\ 
\ 




60 




i 


21.5° 






~1V 














\ 






50 














\ 








13.1° 








"^ 


— ?> 



Fig. 68. 



Substance 


tc, Degrees C. 


Pc Atmospheres 


Water 


365.0* 
130.0 
197.0 
155.4 

30.92 
277.7 
-146.0 
-118.0 
-220.0 
-140.0 


200.5 
115.0 
35.77 
78.9 
77.0 
78.1 
35.0 
50.0 
20.0 
30.0 


Ammonia 

Ether 


Sulphur dioxide 

Carbon dioxide 

Carbon disulphide 

Nitrogen 

Oxygen 

Hydrogen 

Air 



* According to the recent experiments of Holborn and Baumann, the critical 
temperature of water is 706.1° F (874.5° C) and the critical pressure is 3200 lb. 
per square inch. See article by Prof. Marks, Jour. A. S. M. E., Vol. 83, p. 563. 



200 SUPERHEATED VAPORS [chap, xi 

It appears from the definition of the critical temperature that 
it is possible for a mixture of liquid and vapor to exist only for 
temperatures below %. At higher temperatures the mass re- 
mains homogeneous throughout the entire range of pressure. 
Although at sufficiently high pressure the fluid may be in the 
liquid state, the closest observation fails to show where the 
gaseous state ceases and the liquid state begins. As stated by 
Andrews, the gaseous and liquid states are to be regal'ded as 
widely separated forms of the same state of aggregation. 

It has been proposed to make the critical temperature the 
basis of a distinction between gases and vapors. Thus, air, 
nitrogen, oxygen, nitric oxide, etc., whose critical temperatures 
are far below ordinary temperature, are designated as gases, 
while steam, chloroform, ether, etc., w^hose critical temperatures 
are above ordinary temperature are designated as vapors. 

The determination of the critical values f^, p^-^ and v^ by ther- 
modynamic principles is a problem of great theoretical interest, 
but lies beyond the scope of this book. 

130. Equations of van der Waals and Clausius. — Many 
attempts have been made to deduce rationally a single charac- 
teristic equation, which with appropriate change of constants 
will represent the properties of various fluids in all states from 
the gaseous condition above the critical temperature to the 
liquid condition. Such a general equation is that of van der 
Waals, namely, 

which was deduced from certain considerations derived from 
the kinetic theory of gases. As van der Waal's equation does 
not accurately represent the results of Andrew's experiments 
on carbon dioxide, Clausius suggested a modification of the 
last term of the equation and ultimately arrived at an equation 
of the form 

^ V - a {v^cy^ ^ ^ 

where /( T) is a function of the absolute temperature that takes 
the value 1 at the critical temperature. 



ART, 131] 



THE MUNICH EXPERIMENTS 



201 



The equations of van der Waals and Clausius are constructed 
with special reference to the behavior of fluids in the vicinity 
of the critical state ; hence they apply more particularly to 
such fluids as carbon dioxide, the critical temperature of which 
is within the range of temperature encountered in the practical 
applications of heat media. The critical temperatures of most 
important fluids, as water, ammonia, and sulphur dioxide are, 
however, far above the ordinary range, and for these media 
the general equations do not give as good results as certain 
purely empirical equations deduced from experiments covering 
a relatively small region. For some fluids, notably ammonia, 
there is unfortunately a lack of experimental data; for the 
most important fluid, water, we have, however, reliable data 
furnished by the recent experiments at Munich. 

131. Experiments of Knoblauch, Linde, and Klebe. — The 

experiments made at the Munich laboratory were so con- 
ducted that three important 
relations could be obtained 
simultaneously. These 
were : 

1. Relation between pres- 
sure and temperature of 
saturated steam. 

2. Relation between spe- 
cific volume and temperature 
of saturated steam. 

3. Relation between pres- 
sure and temperature of 
superheated steam with the 
volume remaining constant. 

The experiment covered 
the range 100° to 180° C. 
The apparatus employed is 
shown diagrammatically in 

Fig. 69. An iron vessel a contains a smaller glass vessel h to 
which is attached a glass tube e. A similar glass tube d leads 
from the outer vessel a^ and the two are connected at h with 




7V- 
To Manometer 




202 SUPERHEATED VAPORS [chap, xi 

a tube /leading to a mercury manometer. Steam is introduced 
into vessel a from a boiler, and suitable provision is made for 
returning the condensed steam to the boiler. 

A given weight of water is put into the glass vessel h and 
is evaporated gradually by the heat absorbed from the steam 
surrounding it. As long as vessel h contains a saturated mix- 
ture, the pressure within h must be the same as that within a, 
since the temperature is the same throughout. Hehce the 
mercury levels m, m in tubes c and d will be at the same height. 
When the water in h is all vaporized and the pressure and 
temperature of the steam in a is further increased, the steam 

in h becomes superheated. While 
the temperature is still the same in 
vessels a and 5, the pressures in the 
two vessels are not equal. This 
may be shown hj the 2^s-diagram 
(Fig. 70). Let point A on the 
saturation curve s^^ denote the state 
of the steam in vessel h just at the 
end of vaporization ; it also repre- 
^ ^^^ ^^ sents the state of the saturated 

steam in the outer vessel a. As 
the temperature rises from t^ to t^ the state of the steam 
in a changes as represented by the curve AC \ that is, the 
steam in a is saturated at the pressure f^. The apparatus 
is so manipulated, however, that the mercury level m in tube c 
is held constant, thus keeping a constant volume of steam in 
vessel h. The point representing the state of the steam in h 
moves along the constant volume curve AB in the superheated 
region, and the final pressure ^3 given by the point B is smaller 
than the pressure 'p^ of the saturated steam in a. As a result 
the mercury level in the tube d will be depressed to the 
level n. ' A comparison of the mercury level in the manometer 
with the level m gives the relation between the pressure and 
temperature of superheated steam at the given constant 
volume V ; and a comparison with the level n gives the 
relation between the pressure and temperature of saturated 
steam. 



ART. 132] EQUATIONS FOR SUPERHEATED STEAM 203 

132. Equations for Superheated Steam. — To represent the 
results of the Munich experiments, Linde deduced the empiri- 
cal equation 



pv = BT- p(\ +.«/?) 



0[^\ -B 



(1) 



In metric units with p in kilogram per square meter^ the con- 
stants have the following values : 

^ = 4T.10 6^=0.031 n=^. 

a = 0.0000002 i)= 0.0052 

With English units and pressures in pounds per square inch^ the 
equation becomes : 

pv = 0.5962 2^-^9(1 + 0.0014^) /'l^M^^^^i _ O.SSsY (2) 

The form of Eq. (1) is such as to make it inconvenient for 
the purpose of computation ; and the constant I) in the last 
term leads to complication in the working out of a general 
theory. A modified form of the equation, namely, 

v-^c=-—--(l-^ctp)— (3) 

is free from these objections and with constants properly chosen 
represents the results of the Munich experiments as accurately 
as Linde's equation. The constants are as follows : 

Metric Units English Units 

B = 47.113 B = 85.87, p in pounds per square foot 

= 0.5963, p in pounds per square inch 
log m = 11.19839 log m = 13.67938 
n = 5 n = 5 

c = 0.0055 c = 0.088 

a = 0.00000085 a = 0.0006, p in pounds per square inch. 

The final equation with constants inserted is therefore 

V + 0.088 = 0.5963 ^ - (1 + 0.0006 j^) ^^^^^f ^^^ • (4) 

This equation is the one that will be used in the subsequent 
developments. 



204 SUPERHEATED VAPORS [chap, xi 

An equation of the simple form 

v-^c=-^^ (5) 

P 

has been proposed by Tumlirz on the strength of Battelli's 
experiments. Linde has shown that this equation may be made 
to represent with fair accuracy the results of the Munich ex- 
periments. For English units and with p in pounds per square 
inch^ the equation becomes 

i; + 0.256 = 0.5962—. (6) 

P 

For moderate pressure this formula is quite accurate, but at 
high pressures and superheat the volumes given by it are con- 
siderably smaller than those indicated by the experiments. 

Two other characteristic equations deserve mention. For 
many years Zeuner's empirical equation 

pv = BT - Cp"" (7) 

has been extensively used. The results of the Munich experi- 
ments have shown that the form of this equation is defective, 
and that it cannot accurately represent the behavior of super- 
heated steam over a wide range. Callendar, from certain theo- 
retical considerations, has deduced the equation, 

, BT ^/273Y .Q. 

^-^ = — -^o(-^j (8) 

which in form resembles Eq. (3), but lacks the factor p in the 
last term. While this equation is somewhat simpler than 
Eq. (3), it is less accurate. 

133. Specific Heat of Superheated Steam. — The experimental 
evidence on the specific heat of superheated steam may be clas- 
sified as follows : 

1. The early experiments of Regnault at a pressure of one 

atmosphere and at temperatures relatively close to 
saturation. 

2. The experiments of Mallard and Le Chatelier, Langen, 

and others at very high temperatures. 



ART. 133] SPECIFIC HEAT OF SUPERHEATED STEAM 205 

3. The experiments of Holborn and Henning at atmospheric 

pressure and at temperatures varying from 110° to 
1100° C. 

4. Recent experiments with steam at various pressures and 

with temperatures close to the saturation limit. Of 
these, the experiments of Knoblauch and Jakob are 
considered the most reliable. 

Regnault concluded from his experiments that at a pressure 
of one atmosphere the specific heat of superheated steam has 
the constant value 0.48 for all temperatures. This value has 
been largely used for all temperatures and for all pressures as 
well. 

Experiments by Mallard and Le Chatelier and by Langen at 
high temperatures agree in making the specific heat a linear 
function of the temperature. Thus, according to Langen, 

c^ = 0.439 + 0.000239 t, (1) 

where t is the temperature on the C. scale. 

The earlier experiments of Holborn and Henning at much 
lower temperatures than those of Langen lead to the formula 

c^ = 0.446 -H 0.0000856 t. (2) 

This is again a linear relation, but the coefficient of t is smaller 
than that in Langen's formula. Equations (1) and (2) show 
that the specific heat varies with the temperature at least, and 
that the convenient assumption of the constant value 0.48 is 
not permissible. 

Finally, the experiments of Knoblauch and Jakob show con- 
clusively that Cp depends also upon the pressure. In these 
experiments, steam was run through a first superheater in 
which all traces of moisture were removed. It was then run 
through a second superheater consisting of coils immersed in 
an oil bath. The heat was applied by means of an electric 
current and could be measured quite accurately, and a com- 
parison of the heat supplied with the rise of the teaiperature of 
the steam gave a means of calculating the mean specific heat over 
the temperature range involved. Experiments were conducted 
at pressures of 2, 4, 6, and 8 kg. per square centimeter. The 



206 SUPERHEATED VAPORS [chap, xi 

results are shown by the points in Fig. 71. From these 
results the following conclusions may be drawn : (1) The 
specific heat varies Avith the pressure, being higher the higher 
the pressure at the same temperature. (2) With the pressure 
constant, the specific heat falls gradually from the saturation 
limit, reaches a minimum value, and then rises again. 

Starting with the characteristic equation (3), Art. 132, it is 
possible to deduce a general equation for the specific heat c^, 
that will give results substantially in accord with the experi- 
mental results of Knoblauch and Jakob. For this purpose we 
make use of the general relation 

From the characteristic equation, 

^H-^ = ^ (l + «i?)^, (4) 

we obtain by successive differentiation 



dv B 



BT p 



- + 



mn 



^^i(l + «p), (5) 



d'^v mn(n 4- 1) .., , . ..,. 



Substituting in (3), the result is 

<¥)• a) 



Taking T as constant and integrating (7) with p as the in- 
dependent variable, the result is 

Cp = -^^ — '-p[l + ^i^J-f const, of mtegration. 

Now since T was taken as constant, the constant of integration 
may be some function of T\ hence we may write 

.,=Ky)+ ^"y^> i>(i+|i>) (8) 

Inspection of (8) shows that as T is increased the last term 
grows smaller ; in fact, c^ approaches (pCT) as 7^ is indefinitely 



ART. 133] SPECIFIC HEAT OF SUPERHEATED STEAM 207 



100 




150 



300 



350 300 

Temperature C. 

Fig. 71. 



350 



400 



208 SUPERHEATED VAPORS [chap, xi 

increased. From Langen's experiments, it is seen that at very 
high temperatures c^ is given by an equation of the form 

hence we are justified in assuming that 

where a and /3 are constants to be determined from experi- 
mental evidence. Equation (8) thus becomes 

., = «+^2'+^^^^|^^(l+|4 (9) 

This is the general equation for the specific heat of superheated 
steam at constant pressure. 

It may be seen at once that this equation gives results agree- 
ing in a general way with those of Knoblauch and Jakob. At 
a given temperature T the specific heat increases with the pres- 
sure ; furthermore for a given pressure, c^ has a minimum value 
as appears by equating to zero the derivative 

The following values of the constants have been found to 
make Eq. (9) fit fairly well the experimental results of Knob- 
lauch and Jakob : 

a = 0.367 

/? = 0.00018 for the C. scale. 

/S= 0.0001 for the F. scale 

Replacing the product Amn(^n -f- 1) by a single constant (7, 
we have as the final formula for the specific heat 

e^ = 0.367 + 0.0001 T-^pQl + 0.0003 p} ^, (10) 

where log 0= 14.42408 (pressure in pounds per square inch). 
Figure 71 shows the curves representing this formula for the 
pressures of the Knoblauch and Jakob experiments. The 
agreement between the points and curves is satisfactory, con- 
sidering the difficulty of the experiments. In Fig. 72 the 
Cp-cnryes for various pressures in pounds per square inch are 



ART. 133] SPECIFIC HEAT OF ' SUPERHEATED STEAM 209 



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Superheat, Hq^. E. 

Fig. 72. 



500 



600 



210 SUPERHEATED VAPORS [chap, xi 

shown. The abscissas are, however, not temperatures but 
degrees of superheat. 

134. Mean Specific Heat. — Formula (10), Art. 133, gives 
the specific heat at a given pressure and temperature. For 
some purposes it is desirable to have the mean specific heat be- 
tween two temperatures, the pressure remaining constant. 
This is readily calculated by the mean value theorem ; thus 
denoting by (^Cp)„^ the mean specific heat, we have 

Using the general expression for c,,, we laave, therefore, 

^ A 1 f^'^r , om , Amn(n-^V) (-, , a Wjm 



Amp(n + l)(l + lp){jrn - ^„)- 



rp n 
f— ' rr" ^ ' ^^^ (2) 



The calculation, while straightforward is rather long, and if 
Cp-curves are available, it is usually preferable to determine 
the mean e^, by Simpson's rule or by the planimeter. 

Curves of mean specific heat are shown in Fig. 73. For any 
degree of superheat the mean specific heat between the satura- 
tion state and the given state is given by the ordinate corre- 
sponding to the given degree of superheat and the given 
pressure. For example, at a pressure of 150 lb. per square 
inch the mean specific heat for 240° superheat is 0.529. 

135. Heat Content. Total Heat. — Having a formula for the 
specific heat at constant pressure, equations for the heat con- 
tent and the intrinsic energy of a unit weight of superheated 
steam at a given pressure and temperature are readily derived. 
For this purpose the general equation 

dq = c^dT- At(~) dp (see Art. 54) (1) 



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212 SUPERHEATED VAPORS [chap, xi 

is most convenient. Since by definition 

i= A(lu -{-pv'), 
we have di = A \_du + d (^v)], 

or di = dq-\- Avdp. (2) 

Hence, making use of (1), 

di = c^dT-A[T^-vyp. (3) 

From the characteristic equation we have 

whence rg,- „ = („+ 1)(1 + »^)^^. 

Introducing in (3) this expression for T -—— v and the general 

expression for Cp^ the result is 

di = (a + fiT)dT-^ Amnin + r)p{\ + \v\-^^ 

- ^^y^^ (1 + «i>) * - ^^^i>. (4) 

Since i depends upon the state of the subtance only, the second 
member of (4) must be an exact differential. The integral is 
readily found to be 

i = «y+|r^-^f« + i)p(i+^;>)^-^«P + v (5) 

The constant of integration i^ is determined by applying 
Eq. (5) to the saturation state. For a given pressure and cor- 
responding saturation temperature the second member of (5) 
exclusive of i^ can be calculated. The first member is the 
value of i for the assumed pressure as given in the steam table. 
Hence i^ is found by subtraction. By this method the mean 
value 2q= 886.7 is obtained. 

Introducing known constants, Eq. (5) becomes 

i = ^(0.367 + 0.00005 2^) - ^ (1 + 0.0003 jt>)-^ 

-0.0163jt?-|- 886.7. (6) 



ART. 135] 



HEAT CONTENT. TOTAL HEAT 



21S 



Here log C = 13.72511 when p is taken in pounds per square 
inch. 

The total heat of a unit weight of superheated vapor is the 
heat required to raise the tem- 
perature of the liquid to the 
boiling point at the given con- 
stant pressure, evaporate it, and 
then superheat it, still at con- 
stant pressure, to the tempera- 
ture under consideration. On 
the TO'-plane, the process is 
shown by the line ABCD (Fig. 
74). The area OABCC^ rep- 
resents the total heat of the 
saturated vapor, which has 
been denoted by q" . The area 

C^CDB. represents the heat added to superheat the vapor. 
This heat is evidently given by the integral 

C 




Fig. 74. 



^c^dT^^ 



a-\-^T 



Jin 



-P 



(-1^)] 



dT 



taken between the saturation temperature T^ at point C and 
the final temperature T at point D. This integral is, in fact, 
the product ((?p)^(^— T^), where (e^)^ is the mean specific 
heat for the temperature range T— T^. The total heat of a 
unit weight of superheated steam is given therefore by the 
expression ^ ^ ^„ ^ (O-C^ - T,). (7) 

The term (^Cp)^(^T — T^) is easily found from the mean 
specific heat curves (Fig. 73), and q"Q=i"^ is given in the 
steam table. Hence Avith the aid of the curves, an approxi- 
mate value for the heat content may be calculated. 

Example. Find the heat content of one pound of steam at a pressure 
of 150 lb. per square inch superheated 200°. 

From the steam table {"(= q") for this pressure is 1194.6 B. t. u. ; and 
from Fig. 73 the mean specific heat from saturation to 200° superheat is 
0.534. 
Hence { = 1194.6 + 200 x 0.534 = 1301.4 B. t. u. 

The result given by formula (6) is 1301.7 B. t. u. 



214 SUPERHEATED VAPORS [chap, xr 

136. Intrinsic Energy. — For the intrinsic energy we have 
from the defining equation ^ = A(u + pv}, 

Au = i — Apv, (1) 

Using the expressions for i and v heretofore derived, we obtain 
the equation 



Au= Tia-hi/BT-AB)-^^'^ 



Jfr 



n+(n—l)-p 



This expression gives the intrinsic energy in B. t. u. of a unit 
weight of superheated steam. Introducing the proper constants, 
we have, when p is taken in pounds per square inch, 

^M = 2^(0.2566 + 0.00005 2^)- ^(1 + 0.00024 jt?) + 886.7, (3) 

where log (7=13.64593. 

The intrinsic energy may also be found quite exactly by 
the following method. For the given pressure p the energy 
of one pound of saturated steam is 

Au" =q' + p, 

and the increase of energy due to the superheat is 

where (<?^)« denotes the mean specific heat at constant volume. 
The difference (c^)^ — (^t,)« varies somewhat with the pressure 
and superheat, but 0.13 may be taken as a mean value. Hence 
the energy of one pound of superheated steam is given by the 
equation 

^» = j'+p+[(c,)„-o.i8](y-2;). (4) 

Values of q^ and p are given in the steam table and the 
proper value of (c^)^ may be found from the curves of Fig. 73. 

Example. Find the intrinsic energy of one pound of steam at a pres- 
sure of 150 lb. per square inch and superheated 100°. 

From the steam table q' = 329.8 B.t.u. and p = 781.6 B.t.u. The value 
of {cp)m is found to be 0.553. Hence we have for the energy of one pound 

Au = 329.8 + 781.6 + (0.553 - 0.13)100 = 1153.7 B.t.u. 

The same result is obtained by using formula (3). 



ART. 137] ENTROPY OF SUPERHEATED VAPORS 215 , 

137. Entropy. — From the general equation 



we have 



dq=c,dT-AT[^^dp. 



ds=^=e,^-A(^dp. (1) 



Introducing in this equation the expressions previously derived 
for Cj, and [ -— J (see Art. 133), the result is 

''' = (1-+ ^)'^^+ ^™"^('* + 1) (i + ^ p) 1^. - ^^* 

-~^0- + <^P')dp- (2) 

This is necessarily an exact differential since s is a function of 
the state only. The integral is found to be 

s = « log, T+ fiT-AB log,p - Anp(l + 1^)^^+ s„. (8) 

Inserting the known constants and passing to common loga- 
rithms, (3) becomes 

s = 0.8451 log T-{- 0.0001 T- 0.2542 log j9 

-^(1 + 0.0003^) |-g- 0.3964. (4) 

In using (4), p is taken in pounds per square ineh^ and 
log (7=13.64593. The constant 0.3964 is determined by 
passing to the saturation limit, as was done in finding the 
value of Zq. 

Equation (4) gives the entropy of one pound of superheated 
steam at any given pressure and temperature. 

The entropy may also be found as follows. Let the point D 
(Fig. 74) represent the state of the fluid and assume CD 
to be a constant pressure line cutting the saturation curve 
at C. Then 00-^ gives the entropy s" of saturated steam 
at the same pressure as the superheated steam, and O^B^ 
gives the increase of entropy in superheating the steam. 



216 SUPERHEATED VAPORS [chap, xi 

Since the superheating is at constant pressure, the increase of 
entropy is 

Hence the entropy of the steam in the state D is 



8= S' 



(5) 



Equation (3) is preferable, however, as it does not contain the 
saturation entropy s'^ 

EXERCISES 

1. Find the entropy, energy, and heat content of 1 lb. of superheated 
steam at a pressure of 80 lb. per square inch absolute and a temperature 
of 410° F. 

2. Saturated steam at a pressure of 140 lb. per square inch absolute is 
superheated to a temperature of 530° F. at constant pressure. Find 
(a) heat added; (h) change of energy; (c) change of entropy per pound 
of steam. 

3. Assume data and compare the results obtained by using (6) and (7) 
of Art. 135. 

4. Assume data and calculate entropy by both (4) and (5), Art. 137. 
Compare results. 

138. special Changes of State. — By means of characteristic 
equation (3), Art. 132, and the general equations that have 
been deduced for the heat content, energy, and entropy, most 
problems that arise in connection vi^ith the changes of state of 
superheated steam may be solved with comparative ease. On 
account, however, of the fact that the specific heat of steam is 
given by a somewhat complicated formula, it cannot be ex- 
pected that the relations here derived will have the simple form 
of those for perfect gases. In the following discussions of 
special changes of state, Ave shall give merely an outline of the 
processes involved, leaving the details to be filled in by the 
student. 



ART. 138] SPECIAL CHANGES OF STATE 217 

1. Constant Pressure. Let superheated steam change state 
at constant pressure from an initial temperature t^ to a final 
temperature t^. For the heat added we have 

The external work is given by the relation 

W=pCv,-v,} = B(T^-TO- mp (l + a|;)[A._^j. (2) 

The change of energy may be found from the energy equation 

u^ — u^ = Jq— W, 

■ i 
or independently by calculating from the general formula the 

energies in the initial and final states. 

The change of entropy may be obtained, likewise, from the 
general equation for entropy or from the relation 

- Amnp(l + Ip^-^i--^)' (^) 

The preceding equations apply to a unit weight of the 
fluid. 

2. Constant Volume. If T^ and ^ denote, as before, the 
initial and final temperatures, respectively, we have from the 
characteristic equation 

— ^-(l + a^2)^' (4) 



C = 



from which p^ may be found. Having 2\, p^^ and T^^ p^., the 
initial and final values of the energy and entropy may be de- 
termined from the general formulas. Since the external work 
is zero, the heat added is equal to the increase of energy. 

3. Isothermal Expansion. Let the initial and final pressures 
Pj and ^2 be given. The final volume v^ follows from the 



218 SUPERHEATED VAPORS [chap, xi 

characteristic equation. For the change of entropy per unit 
weight we have from the general equation for entropy 

s,-s, = ABlog.f^ + ^[p,{l + ^^p,)-p,{l+lp,)-\. (5) 

The heat added during the expansion per unit weight is 
therefore 



q=TCs,-s,^=ABT\og,Pl 

Amn 



r£% 



i'.ll+iPxl-Hl + l 



(6) 



_ m 
^2 — i^i — — 



For the external work, taking dv from the characteristic equa- 
tion, we have 

= £2'log^ + |^(^i2-|>,^). (7) 

The change of energy may be found by combining (6) and (7) 
or from the general equation of energy. It is found to be 

(^l-i'2)+|(»-l)(l'l'-^2')]- (8) 

It should be noted that in the case of superheated steam con- 
stant temperature does not^ as with perfect gases, indicate con- 
stant intrinsic energy. 

4. Adiahatic Change of State. For an adiabatic change the 
entropy remains constant ; hence, for the relation between the 
final pressure p^ and temperature T^^^ we have from the general 
equation for entropy 

« log, T^ + ^T^-AB log,p2 - Anp^ (l + Ip^-^, = O, 

where (7 is a constant determined from the initial state. The 
pressure p^. i^ generally given; therefore, we have the tran- 
scendental equation 

«log, r, + pT^-p.^{l + Ip^f^, = 0+AB\og,p^ = C", (9) 
from which the value of T^ may be found by successive trials. 



ART. 138] SPECIAL CHANGES OF STATE 219 

Having the initial and final values of p and T^ the initial and 
final values u-^ and u^ of the intrinsic energy may be calculated. 
The external work per unit weight is then 

W=u^-u^^. (10) 

In problems connected with the flow of steam the change of 
heat content resulting from an adiabatic expansion is required. 
This difference is found by calculating from the general equation 
for the heat content the initial and final values ^J and i^. 

If the adiabatic expansion is carried far enough, the expansion 
line, as DE (Fig. 74), will cross the saturation curve s^'^ and the 
state-point will enter the region between the curves s' and s". 
This means that at the end of the expansion the fluid is a mix- 
ture of liquid and vapor. The investigation of this case presents 
no difficulties. The entropy and energy at the initial point I) 
are calculated from the general equation. Knowing the pressure 
for the final state E^ the quality x is readily determined from 
the equation 

«i = V + ^> (11) 

where s^ denotes the entropy in the initial state. Having x, the 
energy in the final state is calculated from the equation 

Then the external work per unit weight is given by the equation 
W= u^-u^== Wj - J(q^' + x^p^}. (18) 

Example. Steam at a pressure of 150 lb. per square inch absolute and 
superheated 100° F. expands adiabatically to a pressure of 5 in. of mercury. 
Required the final condition of the fluid and the external work per pound ; 
also the pressure at which the steam becomes saturated. 

From the general equation the entropy in the initial state is found to be 
1.6346. From the steam table we obtain for the final pressure s' — 0.1880, 

-= 1.7170; hence 

^ 1.6346 = 0.1880 + 1.7J70X, 

or x = 0.8425. 

In the initial state the energy in B. t. u. is 

Aux = 918.1(0.2566 + 0.00005 x 918.1) _ J-|^(l + 0.00024 x 150) + 886.7 
= 1153.9 B. t. u. 



220 SUPERHEATED VAPORS [chap, xi 

In the final state the energy is 

Au2 = qoj + X2P-2 = 101.7 + 0.8425 x 953.7 = 905.2. 

Hence, the external work per pound of steam is 

W=ui-U2= 778(1153.9 - 905.2) = 193,490 ft.-lb. 

The initial entropy 1.6346 is the entropy of saturated steam at a pressure of 
66.6 lb. per square inch. Hence the steam becomes saturated at this pressure. 

139. Approximate Equations for Adiabatic Change of State. — 

Exact calculations that involve adiabatic changes of superheated 
steam are tedious on account of the transcendental form of the 
equation for entropy ; and it is therefore desirable to introduce 
simplifying approximations, provided the results obtained by 
them are sufficiently accurate. An investigation of a number 
of cases covering the range of values ordinarily used in the 
technical applications of superheated steam shows that a set of 
equations similar in form to the equations for a perfect gas 
may be obtained, and that the error involved in using these 
approximate equations does not in general exceed one or two 
per cent. 

The relation between pressure and volume during an adiabatic 
change may be represented approximately by the equation 

p (^v -{- c')"' = const. (1) 

The value of c is taken the same as in formula (4), Art. 131, 
namely, c = 0.088. 

The value of n probably varies slightly with the initial pres- 
sure and w^th the degree of superheat ; however, it appears that 
the value w = 1.31 gives quite accurate results for the range of 
pressure and superheat found in practice. If now we take the 
approximate characteristic equation 

p(y^c^ = BT, (Art. 132) (2) 

we get by combining (1) and (2), 

T = ^P\ • (3) 

n-l 



or 






ART. 140] TABLES FOR SUPERHEATED STEAM 221 

For the external work, we have 

w= rpdv= r ^"^ ^^= yi(''i + o-;>2(''2 + o . (5) 

Given the initial state of the fluid, the volume in the final 
state may be found from (1), the final temperature from (4), 
and the external work from (5). 

Example. A pound of superheated steam at a pressure of 200 lb. per 
square inch and superheated 200° expands adiabatically to a pressure of 
50 lb. per square inch. Required the final condition and the external work. 

The initial volume is found to be 2,973 cu. ft., and the initial entropy 
1.6657. Using the formula for s (Art. 137), the final temperature is found 
by trial to be 752.5° absolute ; and taking this value of- 7^2, the exact value 
of the final volume is found to be 8.681 cu. ft. 

From (3), Art. 136, the energy in the initial state is found to be 1200.57 
B. t. u., that in the final state 1098.82, B. t. u. ; hence the external work is 
778 (1200.57 - 1098.82) = 79,262 ft.-lb. 

Taking the approximate formulas, we have 

1 j_ 

v^ + c= (I'l + c) (^y'^= (2.973 + 0.088) (^V'' = 8.819; 

whence v^ = 8.819 - 0.088 = 8.731 cu. ft. 

To =Ti(^] •" = 1041.4 (—] '" = 749.8°. 

j^ = Pi(vi + c)-p2(v2+c) ^ M4 ,^00 X 3.061 - 50 x 8.819) = 79,550 ft.-lb. 
n-1 0.31^ ^ 

It will be seen that for practical purposes the results obtained from the 
approximate equations are satisfactory as regards accuracy. 

140. Tables and Diagrams for Superheated Steam. — The lead- 
ing properties of superheated steam — volume, entropy, and 
total heat — for various pressures and degrees of superheat 
have been calculated and tabulated by Marks and Davis and 
by Peabody. The values in the Marks and Davis tables are 
derived from specific heat curves that differ somewhat from the 
curves of Fig. 72, and they therefore differ from the values 
obtained from the equations of Arts. 135-137. However, 
throughout the range of ordinary practice, the difference does 
not exceed one half of one per cent. 

The Marks and Davis tables are accompanied by graphical 
charts that may be used to great advantage in the approximate 



222 



SUPERHEATED VAPORS 



CHAP. XI 



solution of numerical problems. The principal chart has the 
heat content i as ordinate and the entropy s as abscissa. The 



\ 


\ 


J 




v 




\ 


















\ 


h 




\f 




\ ) 






















h 




j^ 




M 


■A ^'^ 












\ 


o 1 

0/ 


\ 




\ 




Y 
















\l 


«/ 


I 




K 




]\ '' 


\ l\ 


\ c? 










V 


v' 




0/ 


^ \ 




/ v^ 






. 










\ 


K^ 


j\ 




\ 




h 






\ ^ 








\ 


\ 






\7 


\i 


v\ 






W 


y 








^ 


N 


\ 




J\ 


, l/j 


k") 






\\( 


\ 










X 


V 


\ 


M 




Y 






^v 


A 












\ 




\\ 




\/ 






y 




\ 














\\ 


s^ 


\/\ 






(\ 




^\ 


s 










1 


%{ 


^ 


r\ 






\\ 




\V) 


\ 












k 


^\ 


|\\ 






^y 




^y\ 


\ 


^^ 




' 








N 


V) 


\\ 




Y 




A^ 


w 


V 
















^ 




X 




\\ 


u 


A 
















^ 




J\\ 




\\ 


)v 


\V 
















\ 




\\ 




\\ 


\\ 


v\\ 




















^ 




A 


\\ 


V 




















\^ 


\X\ 


\\ 


\\ 


\\ 






g-5 



Fig. 75. 



general character of the chart is shown in Fig. 75. The curve 
marked " saturation curve " represents the condition of dry 



ART. 140] MOLLIER'S CHART 223 

saturated steam at various pressures. The region above this 
curve is the region of superheat, and the lines running approxi- 
mately parallel to the saturation curve are lines of constant 
degree of superheat. Below the saturation curve is the region 
of wet steam, and the lines running parallel to the saturation 
curve are lines of constant quality. The lines that cross the 
saturation curve obliquely are lines of constant pressure. 

The first conception of the heat content-entropy chart is 
due to Dr. R. MoUier of Dresden, hence we shall refer to it as 
the MoUier chart. In addition to the chart published by 
Marks and Davis, one is contained in Stodola's Steam Turbines 
and one in Thomas' Steam Turbines. In the light of the 
recently acquired knowledge of the properties of saturated and 
superheated steam, the Marks and Davis chart must be regarded 
as the most accurate. 

The Mollier chart may be used for the approximate solution 
of many problems that involve the properties of saturated and 
superheated steam, and it is specially valuable in problems on 
the flow of steam. The following examples illustrate some of 
the uses of the chart : 

Ex. 1. Steam at a pressure of 150 lb. per square inch superheated 200° F. 
expands adiabatically to a pressure of 3 lb. per square inch. 

The point representing the initial condition lies at the intersection of the 
constant-pressure line marked 150 and the line of 200° superheat. Locating 
this point on the chart, it is found at the intersection of the hues i = 1300 
and s = 1.687. The heat content and entropy in the initial state are thus 
determined. The line s = 1.687 intersects the constant-pressure curve p = ^ 
on the line { = 1002 ; hence the heat content after adiabatic expansion is 
1002 B. t. u. The quality in the final state is found to be 0.88. 

Ex. 2. When steam is wire-drawn by flowing through a valve from a 
region of higher pressure jo^ to a region of lower pressure p2, the heat content 
remains constant. Steam at a pressure of 200 lb. per square inch and 
quality 0.95 flows into the atmosphere ; required the final condition of the 
steam. 

Drawing a line of constant-heat content from the initial point to the 
curve p = 14.7, it is found that the final point lies above the saturation curve 
and that the steam is superheated about 12° at exit. The entropy increases 
from s = 1.498 to s = 1.766. 

141. Superheated Ammonia and Sulphur Dioxide. — Experi- 
mental evidence regarding the properties of superheated vapors 



224 SUPERHEATED VAPORS [chap, xi 

other than that of water is very scant, and our knowledge of 
such properties is accordingly imperfect. For superheated 
ammonia Ledoux has proposed the characteristic equation 

pv = BT- Cp^, (1) 

and this form has been accepted by Peabody, who derives the 
following values of the constants (English units) : 

^=99, C=710, m = l. 

For sulphur dioxide Peabody uses the same equation with the 
constants : 

j5=26.4, (7=184, m = 0.22. 

According to Regnault the specific heat of superheated ammo- 
nia has the constant value 0.52. It is very likely that this 
specific heat is no more constant than that of superheated 
steam and that it varies with pressure and temperature. How- 
ever, experimental evidence on this point is lacking. Lorenz 
finds that for superheated sulphur dioxide Cp= 0.329. 

The problem that most frequently arises in connection with 
the use of these fluids as refrigerating media is the determi- 
nation of the state of the superheated vapor after adiabatic 
compression. It may be assumed that the relation between 
pressures and temperatures for an adiabatic change follows 
approximately the law for perfect gases, namely: 

ii)=(r- 

Zeuner found that for superheated steam the exponent 

n 

in (2) is equal to the exponent m in the characteristic equation 
(1). Hence, using the values of m assumed by Peabody, we 
have : 

For ammonia n = - =- ——- = 1.333. 

1 — m 1 — 0.25 

For sulphur dioxide n = — — = 1.282. 

J. — u. zz 

Regnault, however, gives for ammonia w = 1.32. 

A second method of finding the temperature T^ at the end 
of adiabatic compression makes use of the properties of the 



n — \ 



ART. 141] 



SUPERHEATED AMMONIA 



225 



saturated vapor. Let A (Fig. 76) represent the initial state, 
and B the final state after adiabatic compression. UA and 
FB are constant-pressure curves. Denoting by TJ the satura- 
tion: temperature correspond- 
ing to the pressure p^, the 
increase of entropy from U 

T 

to A is tfplog^-J^, and the 

total entropy in the state A is 



s/' 4- Cp log. 



TJ 



Likewise, the entropy in the 
state B is 

TJ'' 



?2'^+^pl0ge 





B. 


T2 








p„t: 


— \F' 






\^ 


T, 


P.X 


\ ^ 




A 




V" 



Fig. 76. 



Since AB is an adiabatic, the entropies at A and B are equal, 
and therefore 



j/' + c^ log, 4r7 = s^" + <^v lo^e -^, • 



(8) 



In this equation s^\ s^\ TJ ^ and TJ' are tabular values corre- 
sponding to the given pressures p^ and p^^ and 7^^ is given. 
Hence, T^ is the only unknown quantity. 



EXERCISES 

1. Calculate by Eq. (2), (4), and (6), respectively, of Art. 132 the vol- 
ume of one pound of superheated steam at a pressure of 180 lb. per square 
inch and a temperature of 430° F. Compare the results. 

2. If the products pv are plotted as ordinates with the pressures p as 
abscissas, show the general form of the isothermals T = C when Eq. (3), 
Art. 132 is used ; when Eq. (6) is used. 

3. For ammonia, Peabody gives the following equations for the latent 
heat of vaporization : r = 540 — 0.8 (t — 32). If at the critical temperature 
r = 0, find ?c for ammonia by means of this formula and compare with the 
value of tc given in Art. 129. Explain the discrepancy. 

4. Following the method of Art. 133, deduce an equation for Cp, using 
the approximate equation (.5), Art. 132 ; also using Callendar's equation (8). 

5. By means of Eq. (3), Art. 132, calculate the specific volume of satu- 
rated steam at the following pressures : 5 in. Hg., 20, 50, 150 lb. per square 

Q 



226 SUPERHEATED VAPORS [chap, xi 

inch. Use the saturation temperatures given in the steam table, and com- 
pare the results with the values of v" given in the table. 

6. Calculate the mean specific heat of superheated steam at a pressure 
of 140 lb. per square inch between saturation and 250"^ superheat. Compare 
the result with the curves of Fig. 73. 

7. Using the mean specific heat curves, Fig. 73, find the heat content 
and energy of one pound of superheated steam at a pressure of 85 lb. per 
square inch and a temperature of 430° F. 

8. A pound of saturated steam at a pressure of 120 lb. per square inch 
is superheated at constant pressure to a temperature of 386° F. Find the 
heat added, the external work, and the increase of energy. 

9. The steam after superheating expands adiabatically until it again be- 
comes saturated. Find the pressure at the end of expansion and the 
external work. 

10. The following empirical equation has been proposed for the value 
of Cp very close to the saturation limit : 

(c^)sat=0.4H--— ^, 

in which tc is the critical temperature, 689° F., and tg is the saturation tem- 
perature corresponding to an assumed pressure. Using the curves of 
Fig. 72, calculate the value C for several assumed pressures, and thus test 
the validity of the formula for these curves. 

11. The following equation has also been proposed for the value of Cp 
at saturation : (c^)gat = a -\- htg. Test this equation, and if it holds good 
within reasonable limits determine the constants a and h. 

12. In the initial state 6.4 cu. ft. of superheated steam has a temperature 
of 420° F. and is at a pressure of 160 lb. per square inch. By the approxi- 
mate equations of Art. 139 find the temperature and volume after adiabatic 
expansion to a pressure of 80 lb. per square inch ; also the work of expansion. 

13. Assume for the initial state of superheated steam jOj = 80 lb. per 
square inch, v^ = 20 cu. ft., t^ = 350° F. Plot the successive pressures and 
volumes for an isothermal expansion to a pressure of 30 lb. per square inch. 
Compare the expansion curve with the isothermal of air under the same 
conditions. 

14. With the data of Ex. 13 find the external^ work, heat added, and 
change of energy (a) for the superheated steam ; (b) for air. 

REFERENCES 

The Critical State. Equations of van der Waals and Clausius 

The literature on these subjects is very extensive. For comprehensive 
discussions, reference may be made to the following works : 
Preston : Theory of Heat, Chap. V, Sections 6 and 7. 



ART. 141] REFERENCES 227 

Zeuner: Technical Thermodynamics (Klein) 2, 202-229. 
Chwolson : Lehrbuch de Physik 3, 791-841. 

Characteristic Equations 

Callendar : Proc. of the Royal Soc. 67, 266. 1900. 

Linde : Mitteilungen iiber Forschungsarbeiten 21, 20, 35. 1905. 

Zeuner : Technical Thermodynamics 2, 223. 

Weyrauch : Grundriss der Warme-Theorie 2, 70, 87. 

Specific Heat of Superheated Steam 

Mallard and Le Chatelier : Annales des Mines 4, 528. 1883. 

Langen : Zeit. d. Ver. deutsch. Ing., 622, 1903. 

Holborn and Henning : AVied. Annalen 18, 739. 1905. 23, 809. 1907. 

Regnault : Mem. Inst, de France 26, 167. 1862. 

Knoblauch und Jakob : Mitteilungungen iiber Forschungsarbeiten 35, 109. 

1906. 
Thomas : Proc. Am. Soc. Mech. Engrs. 29, 633. 1907. 

A most complete discussion of the work of various investigators is given 
by Dr. H. N. Davis, Proc. Am. Acad, of Arts and Sciences 45, 267. 1910. 

General Theory of Superheated Vapors 

Callendar : Proc. of the Royal Soc. 67, 266. 
Weyrauch : Grundriss der Warme-Theorie 2, 117. 
Zeuner : Technical 'I^hermodynamics 2, 243. 



CHAPTER XII 

MIXTURES OF GASES AND VAPORS 

142. Moisture in the Atmosphere. — Because of evaporation 

of water from the earth's surface, atmospheric air always con- 
tains a certain amount of water vapor mixed with it. The 
weight of the vapor relative to the weight of the air is slight 
even when the vapor is saturated. Nevertheless, the moisture 
in air influences in a considerable degree the performance of 
air compressors, air refrigerating machines, and internal com- 
bustion motors ; and in an accurate investigation of these ma- 
chines the medium must be considered not dry air but rather a 
mixture of air and vapor. The study of air and vapor mixtures 
is also important in meteorology and especially in problems 
relating to heating and ventilation. Finally, it has been pro- 
posed to use a mixture of air with high-pressure steam as the 
working medium for heat engines, and the analysis of the action 
of an engine working under this condition demands a special 
investigation of air and steam mixtures. 

Experiment has shown that Dalton's law holds good within 
permissible limits for a mixture of gas and vapor. The gas has 
the pressure jo' that it would have if the vapor were not present, 
and the vapor has the pressure p" that it would have if the gas 
were not present. The pressure of the mixture is 

P=p'+p". (1) 

If the vapor is saturated, the temperature t of the mixture must 
be the saturation temperature corresponding to the pressure 
Jt?'^ If the temperature is higher than this, the vapor must be 
superheated. 

The water vapor in the atmosphere is usually superheated. 
Let point J., Fig. 77, represent the state of the vapor, and let 
AB be a constant pressure curve cutting the saturation curve 

228 



ART. 142] 



MOISTURE IN THE ATMOSPHERE 



229 



at B, Further, let m denote the weight per cubic foot of tlie 
vapor in the state A^ and m^, the weight per cubic foot of satu- 
rated vapor at the same temperature, that is, in the state 0. 




The ratio — is called the humidity of the air under the given 

conditions. If the mixture of air and vapor is cooled at constant 
pressure, the vapor will follow the 
path AB and at B it will become 
saturated. Upon further cooling 
some of the vapor will condense. 
The temperature T^ at which con- 
densation begins is called the dew 
point corresponding to the state A. 

The humidity may be expressed 
approximately in terms of pressures. 
Let pj' denote the pressure of the 
vapor in the state A and pj' the 
pressure of saturated vapor at the 

same temperature, hence in the state represented by C. At the 
low pressures under consideration we may assume that the vapor 
follows the gas Iaw pV= MBT. Hence, taking V= 1, we have 

pj' = p^" = mBT, - ^'o ^^ ^o 
and ^ pJ'=m^BT. / 

Therefore, denoting the humidity by (^, we have 



Fig. 77. 



-'o^To 



m p^" 



m 



(2) 



That is, the humidity is the ratio of the pressure corresponding 
to the dew point to the saturation pressure corresponding to 
the temperature of the mixture. 

For investigations that involve hygrometric conditions, the 
data ordinarily required may be found in table II, page 319. 
This table gives the more important properties of water vapor 
at low temperatures. 

Example 1. Air is at 70"" and the dew point is found to be 52"^ F. 
Required the humidity and the weight of vapor in a cubic foot of the 
mixture. 



230 MIXTURES OF GASES AND VAPORS [chap, xii 

/ At 70° the saturation pressure is, from table II, 0.738 inches of Hg, 
while at 52° the saturation pressure is 0.3905 inches of Hg. The humidity- 
is therefore 

0,390_5^,52, 
0.738 

If the air were saturated at 70°, it would contain 8.017 grains of vapor per 
cubic foot. Hence with 52.9 per cent humidity the weight of vapor per 
cubic foot is 

8.017 X 0.529 = 4,241 grains. 

Example 2. Atmospheric air has a temperature of 90° F. and a humidity 
of 80 per cent. It is required that air be furnished to a building at 70° F. 
and with 40 per cent humidity. 

From table II, the pressure of saturated vapor at 70° is 0.738 inches 
of Hg ; hence from (2) the pressure' corresponding to the dew point is 
0.40 X 0.738 = 0.2952 inches of Hg, and the dew point is 44.5°. In the initial 
state one cubic foot of air contains 0.80 x 14.85 = 11.88 grains of vapor. 
The air is cooled to 44.5° by proper refrigerating apparatus and in this state 

contains 3.39 x ^59.6 + 44.5 ^ 3 -j^^ grains, the difference 11.88 - 3.11 = 8.77 

459.6 + 90 ^ 

grains being condensed. The air freed from the condensed vapor is now 
heated to the required temperature, 70°. 

143. Constants for Moist Air. — The constants B^ c^, <?^, 
etc., given in Chapter VII apply only to dry air. For air 
containing water vapor the constants must be changed some- 
what, the magnitude of the change depending, of course, upon 
the relative weight of vapor present. 

An expression for the constant B of the mixture may be 
obtained by the following method. Let the volume V contain 
M^ lb. of air at the pressure p^ and M^^ lb. of water vapor at 
the pressure p" . Then assuming that the gas law may be 
applied to the vapor, we have 

p'V=M,B,Z (1) 

fV=M^B,T, (2) 

whence ^ = — ^ • —?• (3) 

Let — ^ = 2, and — 3 = e ; then from (3) 
M^ B^ 

P p — p' 



ART. 143] CONSTANTS FOR MOIST AIR 231 

whence »'' = p , p^ — p (5) 

Adding the members of (1) and (2), we obtain 

The constant B^ of the mixture is, however, given by the 
equation 

pV={M^ + M^)B^T. (7) 

Hence, comparing (6) and (7), we have 

Taking the molecular weight of water vapor as 18, we have 
^=^=85.72, 

and , = |2=P4? = 1.61. 

B^ t>3.34 

Example. Find the value of B for air at 90° F. completely saturated 
with water vapor. The pressure of the mixture is 14.7 lb. per square inch. 

From the table the pressure p" of the vapor is 0.691 lb. per square inch; 
therefore the pressure j^' of the air is 14.7 — 0.691 = 14.009 lb. per square 

inch. From (5), 1 Jr ez = ^ = ^^^ = 1.0493, ez = 0.0493, and z = ^i^^ 
^ ^ p' 14.009 ' 1.61 

= 0.0306. Therefore, 5^ = 53.34 x i^^^ = 54.31. 

1.0306 

The specific heat of the mixture is found by applying the 
law deduced in Art. 83. If cj and ej' denote respectively 
the specific heats of the air and steam, then the specific heat of 
the mixture is given by the equation 

i-\-z 

Example. Taking Cp for air as 0.24, and for steam at 90° as 0.43, the 
specific hsat of the mixture given in the preceding example is 

0.24 + 0.0306 X 0.43 



1 + 0.0306 



0.2456. 



232 MIXTURES OF GASES AND VAPORS [chap, xii 

144. Mixture of Wet Steam and Air. — In a given volume V 
let there be M^ lb. of air and M^ ^^' of saturated vapor mixture 
of quality x. The absolute temperature of the entire mixture 
is T, and the total pressure p. The pressure p is the sum of 
the partial pressures p' and p" of the air and steam, respec- 
tively. This follows from Dalton's law, which whithin reason- 
able limits holds good for the case under consideration. We 
have then 

p' + p"=p. ' (1) 

p'V=M^BT, C2) 

V=M^lx(y" -v'^ + v'-], (3) 

where, as usual, v" and v' denote, respectively, the specific 
volumes of steam and water at the saturation temperature T. 
The energy of the mixture is the sum of the energies of the 
two constituents ; hence, we have 

AU=M^c^T^-M^{q + xp) + U^. (4) 

Likewise, the entropy of the mixture is 

*S' = Jfi[^,log,y+(<;,-01og.F] + Jf2(«' + f) + '^o- (5) 

By means of these equations various changes of state may be 
investigated. 

145. Isothermal Change of State. — Since 2^ remains constant, 
we have from (4) 

AiU^-U{) = 3£,p<:x^-x{), (1) 

and from (5) 

8^~S,= M,AB log. ^ + Mj-ix, - x{). (2) 

Hence, the heat added is given by the equation 

Q = TiS, - S\) ^ M.ABTlog, ^ + M,r(x, - x,-). 
The external work is 

Tf = J<i - (U, - U{) = M.BTiog,'^ + JM^ir- pX^, - x,) 

= Pi'V,\og,^ + p,"(V,-V,). (4) 



(3) 



ART. 146] ADIABATIC CHANGE OF STATE 233 

In the final state, we have 

or neglecting the small water volume v', 

(5) 

(6) 



^2 = 


: M^Xy, 


hile in the initial state 


r,= 


: M^X^V'', 


ence, combining (5) 


and (6), 


^2 = 


^h 



O) 

From (7) it appears that isothermal expansion is accompanied 
by an increase of the quality rr, that is, by evaporation, while 
isothermal compression involves condensation. 

146. Adiabatic Change of State. — In the case of an adiabatic 
change the final total pressure p^ i^ usually given. Assuming 
that the steam in the mixture does not become superheated, 
the final temperature Tg of the mixture must be the saturation 
temperature corresponding to the partial pressure p^^ of the 
steam. The determination of the final state of the mixture 
involves the determination of two unknown quantities ; namely, 
the partial pressure p^" and the quality x^ of the saturated 
vapor. Hence two relations are required. One is given by 
the condition that the entropy of the mixture shall remain 
constant during the change, the other by the condition that 
the final volume V^ may be considered as occupied by each 
constituent of the mixture independently of the other. 

In the application of the first condition it is convenient to 
use an expression for the entropy of the mixture of a form 
different from that given by (5), Art. 144. In terms of the 
temperature and pressure, the entropy of a unit weight of air 
is given by the expression 

s = ^^log, T-ABlog.p-hs^; 

hence for the mixture we have 

,S'= M,(e,\og, T-AB log.jo') + mJ,' +^ + S,. (1) 



234 MIXTURES OF GASES AND VAPORS [chap, xii 

As the constant jSq disappears when the difference of entropy 
between two states is taken, it may be ignored in the calculation. 
Let jSj^ denote the entropy in the initial state. Then since 
the entropy remains constant, we have 

S, = M, (., log, T, - AB log, ^,') + mIs^ + i?|i) ■ (2) 

In Eq. (2), S-^^ M^, M^^ and the coefficients e^ and AB are 
known, as is the final total pressure p^. The partial pressures 
j>^ and p^' ^ the quality x*^, and temperature T^ are unknown. 
However, T^ depends upon p^' ^ and p^ is found from the 
relation p^ -f p^' = p^ when p^'^ is determined. Denoting the 
final volume by F^, we have 

whence ^9= ^^ , „ ' (^) 

Inserting this expression for x^^ in (2), we have finally 

8, = M, (., log, T, - AB log, ft') + i(fi V + ^ -^) • (4) 

In this equation p^ is the only unknown. The solution is 
most easily effected by assuming several values of p<^' and 
calculating for these the values of the second member. These 
calculated values are then plotted as ordinates with the corre- 
sponding values of p<^' as abscissas and the intersection of the 
curve thus obtained with the line S-^ = const, gives the desired 
value oi p^' . 

The external work of expansion or compression is equal to 
the change of energy. Hence, using the general expression 
for the energy of the mixture, we have 

Example, In a compressor cylinder suppose water to be injected at the 
beginning of compression in such a manner that the weight of water and 
water vapor is just equal to the weight of the air. Let the pressure of the 
mixture be normal atmospheric pressure 29.92 in. of mercury, and let the 
temperature be 79.1° F. The mixture is compressed to a pressure of 120 lb. 
per square inch absolute. Required the final state of the mixture and the 
work of compression per pound of air. 



ART. 146] ADIABATIC CHANGE OF STATE 235 

From tlie steam table the partial pressure of the water vapor correspond- 
ing to 79.1° is 1 in. Hg, hence the partial pressure of the air is 28.92 in. Hg. 
The initial quality x^ is found from the relation 

Vi = Ml^Il = M2xivi", 
Pi' 

, MxBTi 53.34x538.7 ^noi/i 

whence xi = - — ^ = = 0.0214. 

Mo^pi'vi" 28.92 X 0.4912 x 144 x 656.7 

The factor 0.4912 x 144 is used to reduce pressure in inches of mercury to 
pounds per square foot. 

For the entropy of the mixture we obtain from (1) (neglecting the con- 
stant So) 

Si = 0.24 loge 538.7 - 0.0686 log^ (28.92 x 0.4912) +0.0916 + 0.0214 x 1.9482 
= 1.4587. 
Since the ratio of the final to the initial pressure of the mixture is -^=^ = 8.2, 

we assume that the pressure p-r' of the vapor after compression will be 
approximately 8 times the initial pressure pi". Hence we assume p2" = 7, 
8, and 9 in. of mercury, respectively, and calculate the corresponding values 
of the second member of (2). Some of the details of the calculation are 
given. 

From Steam Table 



Pa" (in.ng)jt)2"-i^ p,' t, T, s,' r, v," 

7 3.43 116.57 146.9 606.5 0.2097 1011.1 104.4 ] 

8 3.92 116.08 152.3 611.9 0.2186 1007.9 92.18 I Data 

9 4.41 115.59 157.1 616.7 0.2265 1005.0 82.57 J 

p," c.logT, ABlogp,' • -^, S 

P2^-2 

7 1.5378 0.3264 0.0308 1.4519] 

8 1.5100 0.3262 0.0349 1.4673 [ Results 

9 1.5418 0.3259 0.0390 1.4814] 

The pressure p2" that gives the value aS^ = 1.4587 lies between 7 and 8 in. Hg 
and by the graphical method or by interpolation we find P2" = 7.44 in. Hg, 
or P2" = 3.65 lb. per square inch. Therefore P2' = 120 - 3.65 = 116.35 lb. 
per square inch. From the steam table the following values are found for 
the pressure />," = 7.44 in. Hg : fg = 149.3, ^2 = 608.9, (72' = 117.3, r2 = 100-9.4, 
po = 942.8, V2" = 99. The final quality is 

^ ^ BT2 ^ 53.34 X 608.9 
^^^ P2'v2" 116.35 X 144 X 99 

The external work per pound of air is 

W= /[0.17(149.3 - 79.1) + 117.3 - 47.2 + 0.0214 x 989.8 - 0.01958 x 942.8] 
= 61566 ft. lb. 



= 0.01958. 



236 MIXTURES OF GASES AND VAPORS [chap, xii 

The volume of the mixture at the end of compression is 

y^BT,^ 53.34x608.9 ^ ^^33^ ^^ ^ 
p^' 116.85 X 144 ' 

and the work of expulsion is therefore 

1.9386 X 120 X 144 =: 33498 ft. lb. 

Hence, the work of compression and expulsion is 95064 ft. lb. 

The effect of injecting water into a compressor cylinder may be shown 
by a comparison of the result just obtained with the work of compressing 
and expelling 1 lb. of dry air under the same conditions. 

The initial volume of 1 lb of air is —^—^ zrvi- = 13.574 cu. ft. 

14.7 X 144 

The final volume after adiabatic compression to 120 lb. per square inch is 



13.574 ( IM V"^ = 3.0296 cu. ft. 



1 

120 
The work of compression is 

5^(14.7 X 13.574 - 120 x 3.0296) = 59044 ft. lb., 

the work of expulsion is 3.0296 x 120 x 144 = 52350 ft. lb., and the sum is 
111394 ft. lb. The effect of water injection is therefore to reduce the 
volume and temperature at the end of compression and the work of com- 
pression and expulsion. The reduction of work in this case is about 17 
per cent. 

147. Mixture of Air with High-pressure Steam. — In the pre- 
ceding articles, we have dealt with mixtures of steam and air 
in which the pressure of the vapor content was small. The 
suggestion has been made that a mixture of air at relatively 
high temperature and pressure mixed with steam either super- 
heated, saturated, or with a slight amount of moisture be used 
as a medium for heat engines. An analysis of the action* of 
such a medium in a motor demands in the first place a discussion 
of the process of mixing, afterwards a discussion of the change 
of state of the mixture. 

Let ifc/j lb. of air compressed to a pressure ^^ and having a 
temperature T^ be mixed with M^ lb. of wet steam having a 
pressure p^ ^^^ quality x^- The temperature T2 of the steam 
is, of course, the saturation temperature corresponding to the 
pressure p^. Let F^ denote the volume of the air and F^ the 
volume of the steam. The mixing is accomplished by discharg- 



ART. 147] MIXTURE OF AIR AND STEAM 237 

ing the air into a receiver which contains steam, or vice versa. 
Since under these conditions the pressure of the mixture can- 
not be raised above the pressure of the constituents, the volume 
of the mixture cannot be taken as the original volume Fj of 
the air. We assume, on the other hand, that the conditions 
are such that the volume of the resulting mixture is the sum 
of the volumes of the constituents ; that is, 

V= V, + F,. (1) 

As a second condition, the internal energy of the mixture is 
equal- to the sum of the energies of the constitutents ; hence 
we have the equation of condition 

U= U, + U^. (2) 

Let T denote the temperature after mixing p' the partial pres- 
sure of the air and p" the partial pressure of the steam. Then, 
provided the steam does not become superheated, the tempera- 
ture T must be the saturation temperature corresponding to the 
pressure p^' , 

The following relations are readily obtained. 

V, = ^^^. (3) 

Pi 

or since the quality X2 is nearly 1, 

V, = M,x,v,'^, (4) 

V= Mi^.^ M.xv'\ (5) 

P 

where x denotes the quality after mixing, and v" is the specific 
volume of steam corresponding to the pressure^". 

U,=^M,e,T,. (6) 

U=M^e,T+M^iq' + xp). (8) 

From (2) we have 

M^e,T + M^iq' + xp) = M,c, ?! + Bl,(iq^ + x^^). (9) 



238 MIXTURES OF GASES AND VAPORS [chap, xii 

From (1), 

Having V calculated from (10), we obtain from (5) 

and this expression for x substituted in (9) gives finally 
M,c,T + M^{q' + ^) = M^c,T, + M^iq^ + x^^y (12) 

In (12) the second member is known from the initial condi- 
tions. In the first member q\ /?, and v" are dependent on T\ 
hence T is the one unknown. As usual, the solution is ob- 
tained by taking various values of T and plotting the resulting 
values of the first member of (12). 

Example. Let 1 lb. of wet steam, quality 0.85, at a pressure of 200 lb. 
per square inch, be mixed with 2 lb. of air at a pressure of 220 lb. per 
square inch and a temperature of 400°. Required the condition of the 
mixture. 

From the data given, the following values are readily found : 

Fi = 2.895 cu. ft. ; Vi = 1.948 ca. f t. ; V = 2.895 + 1.948 = 4.843 cu. ft. 

U=Ui-\- C72 = 1273.8 B.t.u. 

Equation (12) becomes 

0.34 T^q' + 4.843 J° = 1273.8. 

v" 

We now assume for p" the values 50, 75, and 100 lb. per square inch; 
from the tables we find the corresponding values of q', p, v", and 2\ and 
calculate the values of the first member. The results are : 

For p" = 50, 981 B. t. u. 

p" = 75, 1222.3 B.t.u. 
p" = 100, 1451 B. t. u. 

Plotting these results, we find p" = 81 lb. per square inch very nearly. The 

temperature of the mixture is therefore 313° F. and the quality of the 

4 843 w 

steam is a: == — — - = 0.897. (5.4 is the specific volume v" corresponding to 
5.4 

a pressure of 81 lb.) The partial pressure p' of the air is found from 

(5) to be 130 lb. per square inch. Hence the pressure of the mixture is 

130 + 81 = 211 lb. per square inch. 



ART. 147] MIXTURE OF AIR AND STEAM 239 

It is seen that, as the result of mixing, the temperature is considerably 
lowered, the pressure takes a value between pi and p2y and the quality of 
the steam is increased. 

If the steam is initially superheated, the preceding equations 
must be modified by inserting for F^ and U^ the appropriate 
expressions for the volume and energy, respectively, of super- 
heated steam. To reduce as far as possible the complication 
of the formulas we shall take the approximate equation (5), 
Art. 132, for the volume. We have then 

V^ = M,v^ = M^ (^ - .). (13) 

The constant B is written with a prime merely to distinguish 
it from the constant for air. The intrinsic energy of the steam 
is given by Eq. (2), Art. 136. This equation can be simplified 
with a small sacrifice of accuracy by dropping the term con- 
taining a. The modified equation then takes the form 

Au=^T(^e+fT)-^^ + ^m.l, (14) 

in which g = 0.2566, /= 0.00005, and log (7= 13.64593. 

From (6) and (14) the energies of the constituents before 
mixing can be calculated, and the sum of these gives the 
energy U of the mixture. We have then as one equation of 
condition 

M^c,T+M^l Tie + fT)-^+ 886.7] = A U. (15) 

Since p" and T are here independent, there are two unknowns 
and a second condition is required. From (3) and (13) the 
initial volumes V^ and V^ are found and the sum gives the 
volume V of the mixture. Then 

B'T 
or y' = _^i_. (16) 

From (15) and (16) the unknowns p'' and I^can be found. 



240 MIXTURES OF GASES AND VAPORS [chap, xii 

Example. Let 5 lb. of air at 60^^ F. be compressed adiabatically from 
atmospheric pressure to a pressure of 200 lb. per square inch and mixed 
with 1 lb. of steam at 200 lb. per square inch superheated 100°. The con- 
dition of the mixture is required. 

The temperature of the air after compression 

r, = 519.6 (||)'^=1095». 

The saturation temperature of steam at 200 lb. per square inch is 381.8° F ; 
hence T^ = 381.8 + 100 + 459.6 = 941.4°. The energy of the air is 

5 X 0.17 X 1095 = 930.75 B. t. u. 

and that of the steam is, from (14), 

941.4 (0.2566 + 0.00005 x 941.4)- C-^^+ 886.7 -: 1160.6 B.t.u. 

941.4^ 

Hence A(U^+U2) = AU = 930.75 + 1160.6 = 2091.35. B. t. u. 

We have then from (15). 

0.85 T + r(0.2566 + 0.00005 T) - Q^ = 1204.65. 

To derive an expression for the partial pressure p" the total volume V must 
be found. Before mixing, the volume of the air is 

y_ M^BT, ^ 5x53.34x1095 ^ ^^^^ ^^ 
^ p^ 144 X 200 ' 

and the volume of the steam is 

F. = ^^ - c = ^:^^^^xMi _ 0.256 = 2.55 cu. ft. 
P2 200 

Hence V= 10.14 + 2.55 = 12.69 cu. ft. 

After mixing the superheated steam at the partial pressure p" and tem- 
perature T occupies this volume ; hence, we have (since M^ = 1) 

„ B'T 0.5962 T 



F+c 12.69 + 0.256 
Introducing this expression for p" in the term —^, that term 'becomes 

C 

— , where log C = 12.30919. The equation in T then becomes 

1.1066 T + 0.00005 7^2 - -^ == 1204.65. 

As the value of T evidently lies between 1000° and 1100°, we assume the 
three values 1000, 1050, 1100 and calculate the first member of this equa- 
tion. The results are : 



AKT. 147] MIXTURE OF AIR AND STEAM 241 



T 


1.1066 T 


0.00005 T^ 




Sum 


1000 


1106.6 


50. 


2.04 


1154.56 


1050 


1161.93 


55.13 


1.68 . 


1215.38 


1100 


1217.26 


60.5 


1.39 


1276.37 



By interpolation it is readily found that T = 1041°. The pressure of the 
steam is 

-" ^'^^^^ ^ ^^^^ = 47.94 lb. per square inch, 







^ ~ 12.946 


while the 


pressure of the air is 


Therefore 


P^ 


53.34 X 1041 X 5 
144 X 12.69 

P=p'+p" 



151.93 lb. per square inch. 

199.9 lb. per square inch. 

The total pressure p should evidently be 200 lb. per square inch ; hence 
the result may be regarded as a check on the calculation. 

Having now the initial condition of the mixture, the condi- 
tion after adiabatic expansion to any assumed lower pressure 
and the work of expansion may be found by the methods of 
Art. 146. 

The discussions of Arts. 146 and 147 furnish the necessary 
equations for the analysis of the action of a motor that uses a^ 
mixture of air and steam as its working fluid. 

EXERCISES 

1. Find the humidity and the weight of vapor per cubic foot when the 
temperature is 85° and the dew point is 70°. 

2. The humidity is 0.60 when the atmospheric temperature is 74° F. 
Find the dew point. 

3. Find the value of B for air at 80° with 70 per cent humidity. Find 
also the specified heat Cp of the mixture. 

4. A mixture of air and wet steam has a volume of 3 cu. ft. and the 
temperature is 240° F. The weight of the air present is 1 lb., that of the 
steam and water 0.4 lb. Find the partial pressures of the air and vapor, the 
total pressure of the mixture, and the quality of the steam. 

5. Let the mixture in Ex. 4 expand isothermally to a volume of 5 cu. ft. 
Find the external work, the heat added, the change of entropy, and the 
change of energy. 

6. Let the mixture expand adiabatically to a volume of 5 cu. ft. Find 
the condition of the mixture after expansion, and the external work. 



242 MIXTURES OF GASES AND VAPORS [chap, xii 

7. Let 1 lb. of steam, quality 0.87, at a pressure of 150 lb. per square 
inch, be mixed with 4 lb. of air at a pressure of 160 lb. per square inch and 
a temperature of 340° F. Find the condition of the mixture. 

8. Let the mixture in Ex. 7 expand adiabatically to a pressure of 40 lb. 
per square inch. Determine the final state of the mixture and calculate the 
work of expansion. 

9. Let 1 lb. of steam at a pressure of 150 lb. per square inch and super- 
heated 140° be mixed with 6 lb. of air at a pressure of 160 lb. per square 
inch and a temperature of 340° F. Find the condition of the mixture. 

10. Let the mixture in Ex. 9 expand adiabatically until the pressure 
drops to 14.7 lb. per square inch. Required the final state of the mixture 
and the work of expansion. 

REFERENCES 

Berry : The Temperature Entropy Diagram, 136. 
Zeuner : Technical Thermodynamics, 320. 
Lorenz : Technische Warmelehre, 366. 



CHAPTER XIII 

THE FLOW OF FLUIDS 

148. Preliminary Statement. — Under the title '' flow of 
fluids " are included all motions of fluids that progress continu- 
ously in one direction, as distinguished from the oscillating 
motions that characterize waves of various kinds. Important 
examples of the flow of elastic fluids are the following : (1) The 
flow in long pipes or mains, as in the transmission of illuminat- 
ing gas or of compressed air. (2) The flow through moving 
channels, as in the centrifugal fan. (3) The flow through 
orifices and tubes or nozzles. The recent development of the 
steam turbine has made especially important a study of the last 
case, namely, the flow of steam through orifices and nozzles, 
and it is with this problem that we shall be chiefly concerned 
in the present chapter. 

Of the early investigators in the field under discussion, 
mention may be made of Daniel Bernoulli (1738), Navier (1829), 
and of de Saint Venant and Wantzel (1839). The latter de- 
duced the rational formulas that to-day lie at the foundation of 
the theory of flow ; they further stated correctly conditions for 
maximum discharge, and advanced certain hypotheses regard- 
ing the pressure in the flowing jet which were at the time dis- 
puted but which have since been proved valid. 

Extensive and important experiments on the flow of air were 
made by Weisbach (1855), Zeuner (1871), Fleigner (1874 and 
1877), and Hirn (1844). These served to verify theory and 
afforded data for the determination of friction coefficients. In 
1897 Zeuner made another series of experiments on the flow of 
air through well-rounded orifices. 

Experiments on the flow of steam were made by Napier 
(1866), Zeuner (1870), Rosenhain (1900), Rateau (1900), 
Gutermuth and Blaess (1902, 1904). 

243 



244 THE FLOW OF FLUIDS [chap, xiii 

Most of the experimental work here noted relates to the 
flow of fluids through simple orifices or through short con- 
vergent tubes. The more complicated relations between veloc- 
ity, pressure, and sectional area that obtain for flow through 
relatively long diverging nozzles have been investigated experi- 
mentally by Stodola, while the theory has been developed by 
H. Lorenz and Prandtl. The flow of steam through turbine 
nozzles has also been discussed by Zeuner. 

149. Assumptions. — In order to simplify the analysis of 
fluid flow and render possible the derivation of fundamental 
equations, certain assumptions and hypotheses must necessarily 
be made. 

1. It is assumed that the fluid particles move in non-inter- 
secting curves — stream lines — which in the case of a prismatic 

channel may be considered paral- 
lel to the axis of tlie channel. 
We may imagine surfaces 
^ „c - stretched across the channel, as 

Fig. 78. ' 

F, F\ F\ etc., Fig. 78, to which 
the stream lines are normal. These are the cross sections 
of the channel. They are not necessarily plane surfaces, but 
they may usually be so assumed with sufficient accuracy. 

2. The fluid, being elastic, is assumed to fill the channel 
completely. From this assumption follows the equation of con- 
tinuity^ namely : 

Fw = Mv, (1) 

in which F denotes the area of cross section, w the mean veloc- 
ity of flow across the section, M the weight of fluid passing in 
a unit of time, and v the specific volume. 

3. It is assumed that the motion is steady. The variables 
jo, V, T giving the state of the fluid and also the mean velocity 
w remain constant at any cross section F ; in other words, these 
variables are independent of the time and depend only upon the 
position of the cross section. 

150. Fundamental Equations. — The general theory of flow of 
elastic fluids is based upon two fundamental equations, which 
are derived by applying the principle of conservation of 




ART. 150] FUNDAMENTAL EQUATIONS 245 

energy to an elementary mass of fluid moving in the tube or 
channel. 

Let Wj denote the velocity v^ith which the fluid crosses a 
section F^ of a horizontal tube, Fig. 79, and w the velocity at 
some second section F. A unit weight of the fluid at section 

F^ has the kinetic energy of motion ^ due to the velocity w^ ; 

hence if u-^ is the intrinsic energy of the fluid at this section, the 

total energy is u^ + ^. Likewise, the energy of a unit weight 

of fluid at section F \^u-\- —- . In general, the total energy at 

section F is different from that at section F^ and the change of 
energy between the sections must arise : (1) from energy 
entering or leaving the fluid in 

the form of heat during the -^ 

passage from ^^ to i^; (2) from | ^^"i 

work done on or by the fluid. ""P 1 1 

The heat enterinsf the fluid per 1 

unit of weight between the two 

sections we will denote by q. Evidently work must be done 
against the frictional resistance between the fluid and tube ; let 
this work per unit weight of fluid be denoted by z. The heat 
equivalent Az necessarily enters the flowing fluid along with 
the heat q from the outside. Aside from the friction work, the 
only source of external work is at the sections F-^ and F. As 
a unit weight of fluid passes section F^^ a unit weight also passes 
section F. Denoting by jo^ and v-^ the pressure and specific 
volume, respectively, at F^^ the work done on a unit weight of 
fluid in forcing it across section F-^ is the product jp^^ ; simi- 
larly, the product fv gives the work done hy a unit weight of 
fluid at section F on the fluid preceding it. For each unit 
weight flowing the net work received at the section F^ and F is, 
therefore, f\^\ ~ P'^- 

Equating the change of energy between i^^ and F to the energy 
received from external sources, we obtain 






pv, 



246 THE FLOW OF FLUIDS [chap, xiii 

or 1-= J"^ + (^^+p^r;^)_(w+j9y). (1) 

This is the first fundamental equation. 

It will be observed that the friction work z drops out of the 
equation ; the effect of friction is to alter the distribution 

2 

between internal energy u and kinetic energy -— at section F^ 
leaving the sum total unchanged. 
Differentiation of (1) gives 

\- du-{- d(pv^ = Jdq^ (2) 

a form of the fundamental equation that is useful in subsequent 
analysis. 

Equation (1), as is apparent, takes account only of initial 
conditions at section F^ and final conditions at section F^ and 
gives no information of anything that occurs between these 
sections. A second fundamental equation taking account of 
internal phenomena between the two sections is derived as fol- 
lows. Consider a lamina of the fluid moving along the channel. 
This element receives from external sources the heat dq and 
also the heat Adz^ the equivalent of the work done against 
frictional resistances. Independently of its motion, the lamina 
of fluid may increase in volume and thereby do external work 
against the surrounding fluid, and its internal energy may 
increase. According to the first law we have, therefore, 

J(dq -h Adz') = du+ pdv. (3) 

The first member represents the energy entering the lamina 
during the passage from F^ to #, du is the increase of energy, 
and pdv the external work done. Combining (3) with (2), we 
get 

"^^vdp + dz^O, (4) 

whence by integration we obtain 

— - — L = - vdp-z, (5) 

2g JPx 



ART. 151] FORMS OF THE FUNDAMENTAL EQUATION 247 

The fandamental equations (1) and (5), or the equivalent dif- 
ferential equations (2) and (4), are perfectly general and hold 
equally well for gases, vapors, and liquids. 

151. Special Forms of the Fundamental Equation. — In nearly 
all cases of flow the heat entering or leaving the fluid is so 
small as to be negligible, and we may, therefore, assume that 
^ = 0. The sum u -\-pv will be recognized as the work equiva- 
lent of the heat content i ; that is, 

u-^pv = Ji. (See Art. 52.) 

Equation (1) of Art. 150 may, therefore, be written in the form 



vP" — w 



•2g 

For a perfect gas 



J(.H-iy (1) 



Tc 
Jl = u-\-pv = j—^P^^ (2) 

Avhence, 

- ^^ = .4tO^:^x-^^0. (B) 

If the fluid is a mixture of liquid and saturated vapor, the 
heat content ^ is practically equal to the total heat. (See 
Art. 86.) Hence we may put 

i= q^ -\- a:r, (4) 

and (1) becomes 



w 



^9 

For a superheated vapor, the general form (1) is used, the 
values of i^ and i being calculated from formula (6), Art. 135. 
Equations (3) and (5) being derived from the first funda- 
mental equation hold equally well for frictionless flow and for 
flow with friction. 

152. Graphical Representation. — A consideration of the 
fundamental equations developed in Art. 150 leads to several 
convenient and instructive graphical representations, in which 
the change of kinetic energy and the effect of friction on this 
change are clearly shown. 



248 



THE FLOW OF FLUIDS 



[chap. XIII 



1. Using p and v as coordinates, let AB^ Fig. 80, be the 
curve representing the relation between pressure and volume 

during the passage of the 
fluid from section F^ to sec- 
tion F. The area ABDC 
between this curve and the 
jt?-axis is given by 

In the case of frictionless 
flow, however, the second 
fundamental equation [(5), 
Art. 150] becomes 




Fig. 80. 



w"- 



W 



1_ — 



9 



— r^ vdp. 



(1) 



Hence for frictionless flow, the increase of kinetic energy is 
given by the area between the jt?-axis and the curve representing 
the expansion. 

2. If the flowing fluid is a saturated vapor of given quality, 
the representation just given applies but the equation of the 
expansion line AB must be 
expressed in the form pv"^ = 
const. It is, therefore, more 
convenient to use the tem- 
perature T and entropy S as 
coordinates. If the flow is 
frictionless and adiabatic, 
the expansion curve AB is 
the vertical isentropic, Fig. 
81. The area OHOAA^ rep- 
resents the total heat of the 
mixture in the initial state A^ 
and the area OHBBA^ the 
total heat in the final state B ; 
hence the difference of these areas, namely, the area ABBC^ 
represents the difference q^ -f- x^r^ — (^q' + xr')^ and from (5), 




Fig. 81. 



ART. 152] GRAPHICAL REPRESENTATION 249 

Art. 151, this area, therefore, represents the increase of energy 

'Up' — tv ^ 
^9 " 
If the initial point is at A' in the superheated region, we 
have 

zi = area OHCAA'A^, 

i = area OHDB'A^, 
^\ - ^ = area A'B'BOAA'. 

3. The work z expended in overcoming friction may be 
shown on either the pv- or the ^/S'-plane. When friction is 
taken into account, the heat Az, the equivalent of the friction 
work 25, reenters the fluid, and consequently the heat content i 
and the volume v are both greater at the lower pressure p 
than they would be were there no friction. Hence the expan- 
sion curve AB'^ Fig. 82 and 83, for flow with friction must 
lie to the right of the curve AB for flow without friction. 
This statement applies to both figures. 

Let ^^ denote the heat content in the initial state J., i the 
heat content in the state B^ and i' the heat content in the final 
state B^ when friction enters into consideration. Then 



whence 



i' > ^, 



^l — i' < ^l — 



It follows from (1) Art. 151, that the change of kinetic energy 
-1- for flow with friction is less than the change 



2^ " 2g 

in the case of frictionless flow. Friction, therefore, causes a 
loss of kinetic energy given by the relation 

= J(i'-i). (2) 



2d 

On the ^xS'-plane, Fig. 83, this loss is represented by the area 
A^BB'B^; for 

i' = area OHBB'B^, 

i = area OHDBB^, 
^BB'B^ 



i' -{='dre^A,BB'B' 



250 THE FLOW OF FLUIDS [chap, xiii 

When, therefore, on account of friction the entropy of the fluid 
is increased by BB' , the area representing the increase of 
kinetic energy is the original area ABDO iov frictionless flow 
minus the rectangular area A^BB'B-^ , The increase of en- 
tropy in this case is due entirely to the heat Az entering the 
fluid ; hence as explained in Art. 50, the increase of entropy 

is ^ r — , and the area AyAB' B^ under the curve AB' repre- 

2 J- 

sents (in heat units) the friction work z. 

On the j92;-plane let a constant i line be drawn from point B\ 

Fig. 82, cutting the frictionless expansion line in the point G-. 

Then since the heat content 
i' is the same at G- as at B'^ 
the difference i-^ — i' in pass- 
ing from A to B' along the 
actual curve is the same as 
in passing from Aio G- along 
the ideal frictionless expan- 
sion curve AB. But the 
change of i between the 
FIG. 82. states represented by points 

A and (r, which in work 

units represents the increase of kinetic energy between A 

and G-^ is given by the area AGEO. Hence we have : 

For frictionless flow, ^ = area ABBO, 

For the actual flow, ^'^ " ^' l = area A QEC. 

Hence the loss of kinetic energy due to friction is given by the 
area BBBG. 

From the fundamental equation (5), Art. 150, we have 




\ vdp — 



fl-wl ^3^ 



^9 
in which the integral refers to the actual expansion curve. 

Referring to Fig. 82, j vdp is given by the area AB'BO 



AKT. 152] 



GRAPHICAL REPRESENTATION 



251 




while the change of energy for the actual flow is, as just shown, 

given by the area ACrUO; hence the difference, the area 

AB'DUGA, represents the work 

of friction z. 

The friction work z (area 

A^AB'B^', Fig. 83) is greater 

than the loss of kinetic energy 

(area A^BB'B^^. The reason 

for this lies in the fact that 

part of the heat Az entering 

the moving fluid is capable of 

being transformed back into 

mechanical energy. As shown 

in Chapter IV, the loss of 

available energy, represented 

by area A^BB'B^^ is the increase of entropy multiplied by the 

lower temperature. The triangular area ABB^ represents, 

therefore, the part of the friction work that is recovered. 

4. The most convenient graphical representation for practi- 
cal purposes is obtained by taking the heat content i and entropy 

s as coordinates. On this is- 
plane a series of constant pres- 
sure lines are drawn, Fig. 84 ; 
then a vertical segment AB 
represents a frictionless adia- 
batic change from pressure p^ 
to a lower pressure p^ while a 
curve AB' between the same 
pressure limits represents an 
expansion with increasing en- 
tropy, that is, one with fric- 
tion. The segment AB^ there- 




FiG. 84. 



fore, represents the increase of jet energy 



w^ — w 



friction, the segment AGr^ the smaller increase 



2^ 



i- without 



W-, 



2g 



with 



friction, and the segment GrB^ the decrease in final kinetic 
energy due to friction. 



252 



THE FLOW OF FLUIDS 



[chap. XIII 



153. Flow through Orifices and Short Tubes. Saint Venant's 
Hypothesis. — Let the elastic fluid flow from a reservoir in 
which the pressure is p-^ through an orifice or short tube, Fig. 
85, into a region in which exists a pressure p^ lower than p^ 
If we take the section F-^ in the reservoir, the velocity w^ will 
be small and may be assumed to be zero. The second section 
F will be taken at the end of the tube, and 
the pressure at this section will be denoted 
hj p. Assuming the flow to be frictionless 
and adiabatic, we have, since w^ = 0, 

j Wp. (1) 




2^ 



The law of the expansion is given by the 
equation 



p^v^'' =pv' 



(2) 



Fig. 85. 



where for air n= k, while for saturated or 
superheated vapor it has a value depending on the conditions 
existing. In any case, n can be determined, at least approxi- 
mately. Making use of (2) to evaluate the definite integral 
of (1), we get 



n-l 



2^ 



Pi' 



■J 



(3) 



If F denotes the area of the orifice or tube, and M the 
weight of fluid discharged per second, the law of continuity is 
expressed by the equation 

Mv = Fw, (4) 

whence eliminating w between (3) and (4), we obtain 



F 



n-l 



From (2), we have 

which substituted in (5) gives 



(5) 



M=F^\2g 



n— 1 v^ 



f-if) 



(6) 



ART. 153] 



SAINT VENANT'S HYPOTHESIS 



253 




If now various values be assigned to the lower pressure 'p 
and the values of w and M be found from (3) and (6), respec- 
tively, the relations be- 
tween ^, w^ and M will be 
as shown in Fig. 86. The 
initial pressure jp-^ is rep- 
resented by the ordinate ^^' ^ 
OQ^ the lower pressure jt? 
by the ordinate OK^ and 
the curve AB represents 
the change of state of the 
moving fluid starting from 
the initial state A. The 
shaded area GrABH rep- 

resents the integral \ vdp and, therefore, the kinetic energy 

of the jet — at the section F, The abscissa HE represents 
the velocity w found from the equation 

w=^2gx area GABR (in ft. lb.), 

while the abscissa HD represents to some chosen scale the 
weight of fluid discharged per second, as found from (4) or 
directly from (6). Inspection of (6) shows that the discharge 
M reduces to zero when p =Pi and also when jt? = 0. It fol- 
lows that the curve G-DO must have the general form shown 
in the figure and that the discharge i)[f must have a maximum 
value for some value of p between ^ = and p = p^- Let 
this value of p be denoted by p^. Evidently from (6), Mis sl 
maximum when 

2 n+1 

is a maximum. Placing the first derivative of this expression 
equal to zero and solving, we find for the ratio — that makes 
Msi maximum _ji_ 

9 \n-l 

i) ■ C^) 



Pm 
Pi 



n H- 



254 



THE FLOW OF FLUIDS 



[chap. XIII 



This ratio is called the critical ratio, andj?^ is called the critical 
value of the lower pressure p. For air, taking n= k = 1.4, this 
ratio is 0.5283 or approximately 0.53; for saturated or slightly 
wet steam, taking n = 1.135, the ratio is 0.5744. 

The question now arises as to the relation between the pres- 
sure p in the jet at section F and the pressure p^ of the region 
into which the jet discharges. If it be assumed that p and p^ 
are always equal, then ^ = when p^ = 0, and from (6) M— 0. 
This can only mean that no fluid can be discharged into a perfect 
vacuum, a result manifestly absurd. It follows that under 
certain conditions, p must be different from p^. Saint Venant 




0.3 , 0.5 

Fig. 87. 



and Wantzel, to whom equations (3) and (6) are due, asserted 
that the discharge into a vacuum must be a maximum and 
advanced the hypothesis that for all values of p^ lower than the 
critical pressure p^ the discharge is the same. We have, there- 
fore, two distinct cases : (1) If p^ is greater than p^, the pres- 
sure p in the jet takes the value |?2, and w and Mare found from 
(3) and (6), respectively. (2) If p^ is equal to or less than 
p^ the pressure p assumes the constant value p^ given by (7), 
and the velocity and discharge remain the same for all values 
of ^2 between p^ =Pm and p^— 0. 

The hypothesis of Saint Yenant has been fully confirmed by 
the experiments of Fleigner, Zeuner, and Gutermuth. Figure 87 
shows the results of Gutermuth's experiments on the flow of 
steam through a short tube with rounded entrance, using dif- 



ART. 154] FORMULAS FOR DISCHARGE 255 

ferent initial pressures p-^. In each case the discharge becomes 
constant when the lower pressure reaches a definite value p^. 

154. Formulas for Discharge. — Since for all values of jOgless 

than p^ the discharge remains constant and the pressure at the 

plane of the orifice or tube takes the value p^^ we may obtain 

P 
the maximum velocity and discharge by substituting for — in 

^1 Pi 

(3) and (6) of Art. 153 the critical value ( ^ ] "^ . The 

resulting equations are : 



1 

9 \^{^ 



and M=F[-^] ^\2g-^^. (2) 

\n+lj ^ ^ n-\-l v^ ^ ^ 

These equations give w and M for p2<Pm l if P2 ^ Pm the ratio 

— must be substituted for — in the ori spinal equations. 
Pi Pi 

By easy transformations (1) and (2) may be given simpler 

forms. The following are some of the well-known formulas 

that have been thus derived. 

1. Fliegners Equation for Air. From the general equation 

Piv^ = BT, 

which applies to the air in the reservoir from which the flow 

proceeds, we have 

Pi ^ Pi 
v^ BT' 

Substituting this expression for P^ in (2), and taking n = h^ 



the result is 

^ Pi 



M=[^^.r'\-,^ 



k-\-lJ lB(k + 1) J 



VTi 



(3) 



Inserting the numerical values of k and B for air, we get in 
English units 

M=O.F>ZF~^. (4) 



256 THE FLOW OF FLUIDS [chap, xin 

This is the equation given by Fliegner as representing the re- 
sults of his experiments on the flow of air from a reservoir into 
the atmosphere. It holds good when the pressure in the reser- 
voir is greater than twice the pressure of the atmosphere. 

When the pressure in the reservoir is less than twice the at- 
mosphere pressure the following empirical equation is given by 
Fliegner : 



M= 1.06i^V-^^^%— ^ 



(5) 

2. Grashof's Equation for Steam. In formula (2), p^ and v^ 
refer to the fluid in the reservoir. If this fluid is saturated 
steam, then p^ and v-^ are connected by an approximate relation 

PiV{-= (7, (6) 

in which for English units, m = 1.0631 and 0= 144 x 484.2. 
From (6) we readily obtain 



V 



2m 



C2m 



and substituting this in (2), the resulting equation is 



m+l 



^"^^^ V(«+i)c™ 

If now we take for steam the value n = 1.135, (7) reduces to 
the simple form Jf= 0.01911 ^^"-^^ (8) 

In this formula, F is taken in square feet and p in pounds per 
square foot. When the area is taken in square inches and the 
pressure in pounds per square inch, (8) becomes 

iHf =0.0165 J^/-97. (9) 

This formula is applicable for values of p^ below the critical 
back pressure p^^. 

3. Bateau's Formula. Rateau has modified the Grashof 
formula and gives the following as more nearly agreeing with 
the results of his experiments : 



ART. 1551 ACOUSTIC VELOCITY 257 

4. Napier s equations. The following simple, though some- 
what inaccurate, equations based upon the experiments of 
Mr. R. D. Napier, are due to Rankine. 

When the pressure in the reservoir exceeds ^ of the back 
pressure 

M^^; ■ (11) 

when it is less than |^ of the back pressure 



^^^^3(^-^^ (12) 

Example. Find the discharge in pounds per minute of saturated steam 
at 100 lb. pressure (absolute) through an orifice having an area of 0.4 sq. in. 
The back pressure is less than the critical pressure, 57 lb. per square inch. 

1. By Grashof s formula 

M = 60 X 0.0165 X 0.4 x lOO^-^? = 34.493 lb. 

2. By Bateau's formula 

M= 60 X 0.4 X 100 n 6.367 - 0.96 x 2) = 34.673 lb. 

1000 ^ ^ 

3. By Napier's formula 

M = ^-^ ^ ^^^ X 60 = 34.286 lb. 
70 

4. The discharge may be found from the two fundamental formulas 

w = V2^77?7^r^ = 223.7 V/^ - «2, 
and AI = ^. 

V 

The critical pressure p,,, is 57.44 lb. per square inch. From the steam table 
(or more conveniently, and with sufficient accuracy, from the is-chart) 
we find : 

ii (for 100 lb.) = 1186.5 B. t. u. 

{^ (for 57.44 lb.) = 1142.7 B. t. u. 
x„, = 0.964. 
r,n = x,„ (vj' - vj) + vj = 7.07 cu. ft. 

Then w = 223.7 V1186.5 - 1142.7 = 1480 ft. per second, 

and M = 60 X — X i^ = 34.89 lb. 

144 7.07 

155. Acoustic Velocity. — Let p and v denote, as usual, the 
pressure and specific volume of a medium in a given state, and 



258 THE FLOW OF FLUIDS [chap, xiir 



c. 



k the ratio -^ of the specific heats. Then the velocity of sound 
in the medium is given by the relation 



w = Vgkpv. (See any textbook in Physics). (I) 

If we denote by p^ the critical back pressure, we have 



Pi 
which combined with the adiabatic equation 

Prn_f^ 
Pi ~ \^r 

gives 



'fAi^r- ® 



(3) 



Combining (2) and (4), we have 

p^v^ _ k-\-\ 

PmVm ~ 2 

The velocity through the orifice is 



^-1=(-^Y\ (4) 

(5) 



^^V^^rv^i^i^i' 



k_ 
k + l 
and by the use of (5) this becomes 

w = ^gkp^v^. (6) 

Comparing (6) with (1), it appears that the maximum velocity 
of flow from a short convergent tube is the same as the velocity of 
sound in the fluid in the state it has at the critical section. 
This result is due to Holtzmann (1861). 

156. The de Laval Nozzle. — The character of the flow through 
a simple orifice depends largely upon the pressure p<^ in the 
region into which the jet passes. There are two cases to be 
discussed : 

1. When jt?2 is equal to or greater than the critical pressure 
p^ given by the ratio 



Pm 
Pi 

2. When p^ is less than p^. 



U + iy 






ART. 166] 



THE DE LAVAL NOZZLE 



259 





In the first case the pressure at the cross section a, Fig. 88, 

as we have seen, takes the value p^ of the surrounding region, 

and, therefore, the jet experiences no change 

of pressure as it passes into the region beyond 

the nozzle. There is no tendency, consequently, 

for the jet to spread laterally, and for some 

distance beyond the orifice it will have practi- 
cally constant cross section. Furthermore, since 

there is no drop in pressure along the axis of 

the jet, the velocity remains practically con- 
stant at successive cross sections. 
This velocity is given by (1)^ 
Art. 151. 

In the second case the pres- 
sure at section a takes the critical value jt?^, which 
is greater than the pressure of the surroundings. 
As a result of the pressure difference p^ — p^^ the 
jet will expand laterally, as shown in Fig. 89. 
Furthermore, along the axis of the jet the pres- 
sure drops continuously from its initial value p^ 

until at same distance from the orifice it attains the pressure 

p^. Hence, due to this pressure drop, the velocity of the jet in 

the direction of the axis will increase 

as successive sections are passed. The 

initial velocity at section a is 



Fig. 88. 




Fig. 90. 



that is, the acoustic velocity. 

The lateral spreading of the jet may 
be prevented by adding to the orifice 
a properly proportional tube, as shown 
in Fig. 90. The orifice and tube to- 
gether constitute a de Laval nozzle. The tube must diverge 
so as to permit the expansion of the fluid required by the drop 
of pressure from p^ at section a to p^ at section h. The area 
of the end section h depends upon the final pressure p^. At 
section a the jet has the acoustic velocity w^ as if the added 
tube were not present. As the jet proceeds along the tube 



260 THE FLOW OF FLUIDS [chap, xiri 

its velocity increases and at the end section h takes the value 
«^2 given by the relation 

The general character of the flow through the de Laval 
nozzle may be seen from the following analysis. 

Assuming frictionless adiabatic flow, the fundamental equa- 
tions (6) and (7), Art. 150, become, respectively, 

du-\-pdv = 0, (2) 

wdw 7 ^o\ 

(4) 

(5) 









9 


— — vajj. 


We have also the 


equation of 


continuity 








Fw 


= if?;, 


from which 


by differentiation 


we obtain 


For perfect 


gases, 


dw dF 

w F 

u 


_ dv 

V 

pv 

7 1 1 



k-V 

while for superheated or saturated vapors, 

pv 
u = — — r • 
n— 1 

Therefore, (2) becomes 

or kpdv -f- vdp = 0, (6) 

■ dv dp 

whence — = — -i- . 

V kp 

Combining this relation with (5), we obtain 

^ + ^ + ^ = 0. (7) 

w F kp ^ 

Now from (3), 

dw qv 7 

— =^-^dp\ 



ART. 156] 



THE DE LAVAL NOZZLE 



261 



hence (7) becomes 

\kp wV ^ F 
By introducing the equation for the acoustic velocity 

(8) may be readily reduced to the form 

1 dp hiiP' 

p dx 



1 dF 



vP" — w^ F dx 



(8) 



(9) 



(10) 



The variable x may be used to denote the distance of a nozzle 
section from some fixed origin, Fig. 90. For vapors, h may be 
replaced by n. 

The nozzle has two distinct parts: the rounded orifice ex- 
tending from to A^ Fig. 91, and the diverging tube extend- 
ing from A to B, As the 
cross sections decrease in 
area from to A^ the deriva- 

dF 

tive — is negative for this — ^ 
dx 

part ; for the diverging part 

from A to B^ — — is positive ; 
dx 

for the throat A it has the 

value zero. The pressure 

drops continuously from 

to B as shown by the curve 

J. , dp . Fig. 91. 

01 pressure ; hence -^ is 

negative throughout. Referring to (10) we have the following 
results : 




For orifice OA^ — - is — 
dx 

For tube AB, ^ is + 
dx 



dp 
dx 



IS 



k'uP' 



IS — 



dp . 



For throat A^ 



dl[ 

dx 



0; 



dx 

hvP' 



IS 



w^—w^ 



W — Z^/ 

hnP' 

'Up' — W^ 



w<w,. 



is -j- ; w>Ws' 



262 THE FLOW OF FLUIDS [cHAt. xm 

Hence the velocity steadily increases until at the throat it 
attains the acoustic velocity ; then in the diverging tube it 
further increases. Inspection of (10) shows that divergence is 
necessary if the velocity w is to exceed the acoustic velocity w^. 

157. Friction in nozzles. In the case of flow through a simple 
orifice or through a short convergent tube with rounded en- 
trance, the friction between the jet and orifice, or tube, is small 
and scarcely demands attention. With the divergent de Laval 
nozzle, on the contrary, the friction may be considerable and 
must be taken into account. As explained in Art. 152, the 

effect of friction is to produce a decrease in the jet energy — - 

at the end section. Referring back to Fig. 83, suppose A to 
denote the initial state of the fluid entering the nozzle, B^ the 
final state at exit, and B the final state that would have been 
attained with frictionless flow ; then the area A^BB'B^' repre- 
sents the increase in the final heat content ^ due to friction and 
it likewise represents the decrease in jet energy at exit. 

Let 2\, 2*2, and ^y denote, respectively, the heat content of the 
fluid in the states represented by the points J., B^ and B'. 
Without friction, we have 



^ = J(.h-id> 



while with friction 



r,=^(^-^v). _^. 

The loss of kinetic energy due to friction is, therefore, 

It is customary to take as a friction coefficient the ratio of the 
loss of energy to the kinetic energy without friction. Denoting 
this ratio by y we have, therefore, 

%9 ti to 



ART. 157] FRICTION IN NOZZLES 263 

Avhence 

^=J-(l-2/)0-j-V,). (2) 

The experiments that have been made on the flow of steam 
through nozzles indicate that the value of y may lie between 
0.08 and 0.20. 

Example. Steam in the dry saturated state flows from a boiler in which 
the pressure is 120 lb. per square inch absolute into a turbine cell in which 
the pressure is 35 lb. absolute. A de Laval nozzle is used, and the value of 
y is 0.12. Find the velocity of the jet, and the loss of kinetic energy ; also 
the final quality of the steam. 

For the given initial state, i — 1190.1 B.t. u. At the end of adiabatic ex- 
pansion to the lower pressure, x% is found to be 0.925, and ^2 is found to be 
1095.8 B. t. u. The exit velocity on the assumption of frictionless flow is, 
therefore, 

w = 223.7 V1190.1 - 1095.8 = 2172 ft. per second, 

while the actual velocity is 



w' = 223.7 V(l - 0.12) (1190.1 - 1095.8) = 2038 ft. per second. 

The loss of kinetic energy is, 

0.12 X 778 X 94.3 = 8804 ft.-lb., 
or in B. t. u., 

0.12 X 94.3 = 11.3 B.t. u. 

This heat is represented by the rectangle A\BB'Bi, Fig. 83. Hence, for the 
quality x%' in the actual final condition B', we have 

X,' -x,= y(kj^J^ = HA :. 0.012; 
r^ 938.4 

and, therefore, x^' = 0.925 + 0.012 = 0.937. 

The effects of friction are : (1) to decrease the velocity of 
flow at a given section ; (2) to increase the specific volume v 
of the fluid passing the section. The latter effect is seen in 
the case of steam in the increased quality or increased degree 
of superheat due to the heat generated through friction reenter- 
ing the moving fluid. From the equation of continuity 

F=M-, (3) 

w 

it appears that the effect of friction is to increase the numera- 
tor V and decrease the denominator w of the fraction of the 



264 



THE FLOW OF FLUIDS 



CHAP. XIII 



second member ; hence for a given discharge M^ the cross sec- 
tion F must be larger the greater the friction, that is, for the 
same lower pressure p^. 

The effect of friction may be viewed from another aspect. 
In Fig. 92, let the curve CMAE represent the pressures along 
the axis of a de Laval nozzle on the assumption of no friction. 
This curve is readily found for a given value of p^ by finding 

for various lower pressures p 
F. J.'e the proper cross section F by 

means of the two equations, 

— — = JU — ^), and F= 

1g "^ w 

Let ^ be a point on the pres- 
sure curve obtained in this 
manner. If now friction is 
taken into account, the sec- 
tion F^ associated with the 
lower pressure p has a larger 
area than the section F calcu- 
lated on the assumption of 
no friction ; therefore, the 
point A is shifted by friction 
to a new position A^ underneath the new section FK Similarly 
the end section F^ must be increased in area to jP/, and the 
point E on the frictionless pressure curve is shifted to a new 
position W . The effect of friction, therefore, is to raise the 
pressure curve as a whole, that is, to increase the pressure at 
any point in the axis of the nozzle. 

158. Design of Nozzles. — The data required in the design of 
a nozzle are the initial and final pressures, the weight of steam 
that must be delivered per hour or per minute, and the coef- 
ficient y. Two cross section areas must be calculated, that at 
the throat, and that at the end of the nozzle. The following 
example illustrates the method. 

Example. Required the dimension of a nozzle to deliver 450 lb. of 
steam per hour, initially dry and saturated, with an initial pressure of 175 
lb. absolute and final pressure of 15 lb. absolute. Let y — 0.13. 




Fig. 92. 



ART. 158] DESIGN OF NOZZLES 265 

The critical pressure in the throat is 175 x 0.57 = 100 lb. approx. Then 
for frictionless adiabatic flow 

/, = 1196.4 B. t. u., 
C (at throat) = qj + a:^7-^ = 298.1 + 0.962 x 888.4 = 1152.9, 
i2 = q.2' + X2r2 = 181.1 + 0.863 x 969.7 = 1017.5, 

i^ — i^ = 43.5 ; i^ — io = 178.9. 

Since the throat is near the entrance, the effect of friction between entrance 
and throat is practically negligible ; hence the velocity at the throat is 

w^ = 223.7 V43^ = 1475 ft. per second. 
Taking account of the loss of energy (y = 0.13), the velocity of exit is 



iV2 = 223.7 Vo.87 x 178.9 = 2791 ft. per second. 

The quality of steam at the throat was found to be 0.962, and that at exit, 
without friction, 0.863. Because of friction, the quality at exit is increased 
by the amount 178.9 x 0.13 -^ 969.7 = 0.024, thus giving a final quality 
0.863 + 0.024 = 0.887. Neglecting the volume y' of a unit weight of water, 
since x is large, the specific volumes at throat and exit are respectively 

4.42 X 0.962 = 4.252 cu. ft. 

and 26.23 x 0,887 = 23.26 cu. ft. 

From the equation of continuity Ftv = Mv, we have, since 

M = _i^^ ^ 0.125 lb. per second, 
60 X 60 ^ 

j.^^ 0.125 X 4.252 ^QQQQ3g^^_^^^ 
1475 
= 0.0519 sq. in. 
as the area of the cross section at the throat. The area at exit is 

p = 0.125 X 23.26 ^ q 001042 sq. ft. = 0.15 sq. in. 
2791 ^ ^ 

If the cross section of the nozzle is made circular, the diameters at throat 
and exit are respectively 

dra = 0.251 in., d^ = 0.437 in. ; 

and taking the taper of the nozzle as 1 to 10, the length of the conical part 
is 

10(0.437 - 0.257) = 1.8 in. 



EXERCISES 

1. Find the weight of air discharged per minute through an orifice 
J inch in diameter from a reservoir in which the pressure is maintained at 
80 lb. per square inch absolute. The air is discharged into the atmosphere. 



266 THE FLOW OF FLUIDS [chap, xiii 

2. Steam at a pressure of 120 lb. per square inch flows through an 
orifice having an area of 0.4 sq. in. into a region in which the pressure is 
55 lb. per square inch. Find (a) the velocity; (6) the weight discharged 
per minute. Compare the results obtained by using Grashof's, Napier's, 
and Rateau's formulas, respectively. 

3. If in Ex. 2 the back pressure is 80 lb. per square inch, what is the 
weight discharged? Assume the steam to be initially dry and saturated. 

4. If for superheated steam the exponent n in the adiabatic equation 

Pm 

pyw = const, is taken as 1.30, find the critical ratio 

^ Pi 

5. A de Laval nozzle is required to deliver 630 lb. of steam per hour. 
The steam is initially dry and saturated at a pressure of 110 lb. per square 
inch and the final pressure is 8 in. of mercury. Find the necessary areas of 
the throat section and end section of the nozzle, assuming frictionless flow. 

6. In Ex. 5 find the areas of the two sections when the loss of kinetic 
energy is 0.15 of the available energy. 

7. Find the area of an orifice that will discharge 1000 lb. of dry steam 
per hour, the initial pressure being 150 lb. per square inch and the back 
pressure 105 lb. per square inch. 

8. In an injector, steam flows through a diverging nozzle into a combin- 
ing chamber in which a partial vacuum is maintained, due to the condensa- 
tion of the steam in a jet of water. If the initial pressure is 80 lb. per 
square inch and the pressure in the combining chamber is 8 lb. per square 
inch, find the velocity of the steam jet. Assume y = 0.08. 

9. Steam at 160 lb. pressure superheated 100° flows through a nozzle 
into a turbine cell in which the pressure is 70 lb per square inch. Find the 
area of the throat of the nozzle for a discharge of 36 lb. per minute. 

10. Let steam at 160 lb. pressure, superheated 100°, expand adiabatically 
without friction. Take values of the back pressure p2 as abscissas, and plot 
curves showing (a) the available drop in heat content ii — h ; (h) the veloc- 
ity of the jet ; (c) the area of cross section required for a discharge of one 
pound per second. 

Suggestion. Find i^ for the following pressures: 140, 120, 100, 80, 
60, 40, 20, 10, 5 lb. per square inch. Then find w from the formula 
w — 223.7 Vi'i — ^2, and the cross section from the equation of continuity. 

11. Steam at 160 lb. pressure superheated 100° is discharged into a 
region in which the pressure is p^ through an orifice having an area of 
0.25 sq. in. Take the values of p^ given in Ex. 10 and plot a curve showing 
the weight discharged for different values of p2- 

12. Show that if the loss of kinetic energy is y per cent of the available 
energy, the decrease in the velocity of the jet is approximately \ y per cent 
of the ideal velocity. 



ART. 158] REFERENCES 267 

REFERENCES 
General Theory of Flow of Fluids 

Zeuner : Technical Thermodynamics 1, 225 ; 2, 153. 
Lorenz : Technische, Warmelehre, 99, 122. 
Weyrauch : Grundriss der Warme-Thebrie 2, 303. 
Peabody: Thermodynamics, 5th ed., 423. 
Stodola • Steam Turbines, 4, 45. 
Rateau : Flow of Steam through Il^ozzles. 

Original Papers giving Experimental Results or Discussions 

Weisbach : Civilingeineur 12, 1, 77. 1866. 

Fliegner: Civilingenieur 20, 13 (1874); 23, 443 (1877). 

De Saint Yenant and Wantzel : Journal de I'Ecole polytechnique 16. 1839. 

Comptes rendus 8, 294 (1839); 17, 140 (1843); 21, 366 (1845). 
Gutermuth : Zeit. des Verein. deutsch. Ing. 48, 75. 1904. 
Emden : Wied. Annallen 69, 433. 1899. 
Lorenz : Zeit. des Verein. deutsch. Ing. 47, 1600. 1903. 
Prandtl and Proell : Zeit. des Verein. deutsch. Ing. 48, 348. 1904. 
Biichner : Zeit. des Verein. deutsch. Ing. 49, 1024. 1904. 
Rateau: Annales des Mines, 1. 1902. 
Rosenhain : Proc. Inst. C. E. 140. 1899. 
Wilson : London Engineering 13. 1872. 





^_=-^=^ 




^^^ 


---^-F-- 


^^^^^^J rp^ 


— ''^^ 




="^^W^ 



CHAPTER XIY 

THROTTLING PROCESSES 

159. Wiredrawing. — The flow of a fluid from a region of 

higher pressure into a region of much lower pressure through 

a valve or constricted passage gives rise to the phenomenon 

known as wiredrawing or throttling. Examples are seen in the 

passage of steam through pressure-reducing valves, in the 

throttling calorimeter, in the passage of ammonia through the 

expansion valve in a refrigerating 

machine, and in the flow through 

ports and valves in the ordinary 

steam engine. Wire-drawing^ is 
Fig 93. 

evidently an irreversible process, 

and as such, is always accompanied by a loss of available 
energy. 

The fluid in the region of higher pressure is moving with a 
velocity w-^^ Fig. 93. As it passes through the orifice into the 
region of lower pressure 'p^^ the velocity increases to w^ ac- 
cording to the general equation for flow, viz : 

^-i^^jii^-i^y (1) 

The increased velocity is not maintained, however, because the 

energy of the jet is dissipated as the fluid passing through the 

orifice enters and mixes w^ith the fluid in the second region. 

If) ^ in) " 

Eddies are produced, and the increase of energy -^-^ — - is re- 

turned to the fluid in the form of heat generated through in- 
ternal friction. Utimately, the velocity w^ is sensibly equal to 
the original velocity w^ ; therefore from (1), we obtain 

H = H-' (^) 

268 



ART. 160] 



LOSS DUE TO THROTTLING 



269 



as the general equation for a wiredrawing process. The 
initial and final points lie, therefore, on a curve of constant heat 
content. 

160. Loss due to Throttling. — Let steam in the initial state 
denoted by point A, Fig. 94, be throttled to a lower pressure, 
the final state being denoted by 
point B on the constant-z curve 
AB. Also let Tq denote the 
lowest available temperature. 
The increase of entropy during 
the change AB is represented 
by A^B^, and this increase multi- 
plied by the lowest available 
temperature Tq gives the loss of 
available energy. Evidently this 
loss is represented by the area 
A^BCB^. 

Example. In a steam engine the pressure is reduced by a throttling 
valve from 160 lb. per square inch to 90 lb. per square inch absolute. The 
initial quality is a; = 0.99 and the absolute back pressure is 4 in. of mercury. 
Required the loss of available energy per pound of steam. 

From the steam table the initial heat content is 1187.2 B. t. u. At a pres- 
sure of 90 lb. the heat content of saturated steam is 1184.5 B. t. u., therefore 
in the second state the steam is superheated! As the degree of superheat is 
evidently small, it may be determined with sufficient accuracy from the 
curves of mean specific heat. At a pressure of 90 lb. the mean specific heat 




near saturation is 0.55 ; hence the superheat is 



1187.2 - 1184.5 
0.55 



5°, nearly. 



The entropy in the second state is the sum of the entropy at saturation, 
1.6107 for a pressure of 90 lb., and the entropy due to superheat, which is 
approximately. 

0.55 log« TiAl = 0.55 loge ^ = 0.0035. 

Hence, s^ = 1.6107 + 0.0035 = 1.6142. The entropy in the initial state 
is 1.5553, and the lowest available temperature Tq corresponding to the 
back pressure 4 in. Hg. is 585.1°. The loss of available energy is therefore 

585.1 (1.6142 - 1.5553) = 34.46 B.t.u. 

If the drop in pressure is small, the following approximate 
method gives a simpler solution. 



270 THROTTLING PROCESSES [chap, xiv 

While the process is irreversible, we may assume that it is 
replaced by a corresponding reversible change with the condi- 
tion that the heat content i remains constant. The general 
equation 

di = Tds 4- Avdp, 
then becomes, 

= Tds + Avdp, 

and approximately we have, therefore, 

i^> = -^i^P, (1) 

in which As is the increase of entropy corresponding to the 
change of pressure Ap. Since A^ is intrinsically negative, it 
follows that As must be positive. Equation (1) may be 
written in the more convenient form 

^^ = -^% (2) 

For perfect gases (^) reduces to the simple form 

As=-^^^. (3) 

For steam having the quality x^ we have 
V — x(^v" — v'') + v'^ 
and Apv = Apx(y" — v') + Apv' ; 

or neglecting the small specific volume v' of the water, 

Apv = x-yjr. 

Eq. (2) therefore takes the form 

Mean values for jo, T^ and -yjr should be taken. 

Example. If in passing into the engine cylinder the pressure of steam 
is reduced by wiredrawing from 125 lb. to 120 lb. per square inch, what is 
the loss of available energy ? The initial value of x is 0.98 and the pressure 
at exhaust is 16 lb. per square inch. 

Taking the two pressures 125 and 120, the following mean values are 
found from the table : 

^ = 122.5, T = 802.4, i/a = 82.5. Also, A/> = - 5. 



ART. 161] THE THROTTLING CALORIMETER 271 

Hence, As = ^'^^ ^ ^^'^ x ^— = 0.00398. 

802.4 122.5 

For Tq we take the temperature corresponding to the 16 lb., namely, 675.9°. 
Therefore the loss of available energy is 

675.9 X 0.00398 = 2.7 B.t.n. approx. 

161. The Throttling Calorimeter. — A valuable application 
of the throttling process is seen in the calorimeter devised by 
Professor Peabody for determining the quality of steam. In 
the operation of the calorimeter steam from the main is led 
into a small vessel in which the pressure is maintained at a 
value slightly above atmospheric pressure. The steam is thus 
wiredrawn in passing through the valve in the pipe that con- 
nects the main and the vessel. For successful operation the 
amount of moisture in the steam must be small so that, as the 
result of throttling, the steam in the vessel is superheated. 

In Fig. 94, let point A represent the state of the steam in 

the main and point B the observed state of the steam in the 

calorimeter ; then 

^^ = ^V (1) 

But ^ 

^A = h' + ^^r (2) 

where i/ and r-^ refer to the pressure p^ in the main ; 

and ij, = i^" + e^(_t^' - t^}, (3) 

where t^ is the observed temperature of the steam in the 
calorimeter, t^ is the saturation temperature corresponding to 
the pressure p^ ^^ ^^^ calorimeter, ^y is the saturation heat 
content corresponding to the pressure jt^g, and c^ is the mean 
specific heat of superheated steam for the temperature range 
t^ — ^2- Combining the preceding equations, we obtain 

^2 "^ ^p\*2 ^2/ ^1 



X= - 



(4) 



Example. The initial pressure of the steam is 140 lb. per square inch, 
the observed pressure in the calorimeter 17 lb. per square inch, and the 
temperature in the calorimeter 258° F. Required the initial quality. 

The temperature of saturated steam at 17 lb. pressure is 219.4° F. ; hence 
the steam in the calorimeter is superheated 258 — 219.4 = 38.6°. From the 
curves of mean specific heat the value 0.477 is found for the pressure 17 lb. 



272 



THROTTLING PROCESSES 



[chap. XIV 



and the degree of surperheat in question ; and from the steam table we have 
iV = 1153, h' = 324.2, ri = 869. Hence, 



1153 + 0.477 X 38.( 



324.2 



= 0.975. 



The MoUier chart, Fig. 75, may be used conveniently in the 
solution of problems that involve the throttling of steam. 
Since the heat content remains constant during a throttling 
process, the points representing the initial and final states lie 
on the same horizontal line. In the preceding example the 
final point is located from the observed superheat 38.6° and 
the observed pressure 17 lb. in the calorimeter. A horizontal 
line dravrn through this point intersects the constant pressure 
line p = 140 lb., and from this point of intersection the quality 
X = 0.975 is read directly. 

162. The Expansion Valve. — In the compression refrigerat- 
ing machine the working fluid after compression is condensed 

and the liquid under the higher pres- 
sure pi is permitted to flovr through 
the so-called expansion valve into coils 
in which exists a much lower pressure 
P2' Let point A^ Fig. 95, on the liquid 
curve represent the initial state of the 
liquid. The point that represents the 
final state must lie at the intersection 
of a constant -i curve through A and 

^j B^ '^ line of constant pressure jOg- Evidently 

Fig. 95. we have 




and 



"1 ' 



2' 2' 



where x^ denotes the quality of the mixture in the final state 
Therefore .- r _ - r , ^ ^ 



-2' 2' 



or 



Xn — 



The increase of entropy (represented by A^B^ is 



As 



1 . -^2' 2 „ ! 



(1) 

(2) 
(3) 



ART. 163] 



THROTTLING CURVES 



273 






and the loss of refrigerating effect due to the expansion valve, 
which is represented by the area A^ GrBB^^ is 

2'2(V-<) 

,' - T^isl - V). (4) 

The following equalities between the areas of Fig. 95 are 

evident : ^^^^ OEAA^ = area OEFBB^, 

area FCrA = area A^ GrBB^. 

163. Throttling Curves. — If steam initially dry and saturated 
be wiredrawn by passing it through a small orifice into a region 
of lower pressure, then, as has been shown, it will be super- 
heated in its final state. 
If the lower pressure p^ 
is varied, the tempera- 
ture ^2 will ^1^^ vary. 



350 



300 



goO 



200 



50 100 150 

Pressure, lb. per sq. in. 
Fig. 96. 



200 



and the successive values 

of jt?2 a^^ h ^m ^^ ^^P" 
resented by a series of 
points lying on a curve. 
By taking various initial 
pressures a series of such 
curves may be obtained. 
Sets of throttling curves 
for water vapor have 
been obtained by Grind- 
ley, Griessmann, Peake, 
and Dodge. The curves deduced from Peake's experiments 
are shown in Fig. 96. Abscissas represent pressures, ordinates, 
temperatures. The curve from which the throttling curves 
start is the curve t =/(j?) that represents the relation between 
the pressure and temperature of saturated steam. 

It was the original purpose of Grindley, Griessmann, and 
Peake to make use of the throttling curves in finding the 
specific heat of superheated steam. The theory upon which 
this determination rests is simple. From Eq. (4), Art. 161, we 
readily obtain 

(1) 



Cr,= 



^1 ' •^l^l ^2 



t' - L 



274 THROTTLING PROCESSES [chap, xiv 

The temperature difference t^ — f^ for any lower pressure p2 ^^ 
the vertical segment between the throttling curve and the satu- 
ration curve and is given directly by the experiment. Hence if 
the initial quality x is known, and if ^y and i^" are accurately 
given by the steam tables, the mean value of c^ is readily cal- 
culated. The results obtained were, however, discordant and 
of no value. The form of Eq. (1) is such that a slight error 
in any of the terms of the numerator of the fraction produces 
a large error in the calculated value of c^. 

The impossibility of deriving consistent values of e^ by the 
method just described led to the belief that Regnault's formula 
for the total heat of saturated steam, hitherto regarded as 
authoritative, must be incorrect. The experiments of Kno- 
blauch and Jakob on the specific heat having appeared. 
Dr. H. N. Davis of Harvard University discerned the possibility 
of reversing the method and deriving by it a new formula for 
total heat. 

164. The Davis Formula for Heat Content. — The method 
employed by Dr. Davis in deriving from the throttling curves 

a formula for the heat content of 
steam may be described as follows: 
Let AD, Fig. 9T, be one of the 
series of throttling curves, and 
AB' the saturation curve. The 
heat content is constant along the 
throttling curve, that is 

z^ = ^^ = ^^ = etc. 




Fig. 97. Let p^ ^^ ^^^ lowcr pressure cor- 

responding to the points B, B\ 
and let A^ denote the temperature difference indicated by the 
segment B' B. If the steam were made to pass from the satura- 
tion state B^ to the superheated state B at the constant pres- 
sure ^2' the heat absorbed during the process would be c^A^, 
Cp denoting the mean specific heat between B' and B. It follows 
that 

that is, ^A — iBt = Cp ^i' 




ART. 165] THE JOULE-THOMSON EFFECT 275 

In a similar manner the differences i^ — i^,^ i^ — ij)„ etc. are ob- 
tained. The result is a relation between the heat content of 
saturated steam at the original pressure p-^ (state A') and the 
values of the saturation heat content for various lower pressures. 
The temperatures corresponding 
to these pressures are now laid 
off on an arbitrarily chosen line 
MW, Fig. 98, and from the points 
J., B', C\ etc., the segments 

etc. are laid off. A curve 
through the points A, B" , 0" , 
B'\ etc. is an isolated segment of 

the curve giving the relation between the heat content i and the 
temperature t. Necessarily only relative values are thus obtained. 
From the individual throttling curves Dr. Davis thus obtained 
twenty-four overlapping segments of the z^-curve, and by 
properly coordinating- these segments he obtained finally a 
smooth curve covering the range 212° to 400° F. The curve 
was found to be represented by the second degree equation 

i=a + 0.3745(^ - 212) - 0.00055 (t - 212)2 . 

and from the experiments of Henning and Joly on the latent 
heat of steam at 212° F., the value of the constant a was found 
to be 1150.4. 

165. The Joule-Thomson Effect — The classical porous plug 
experiments of Joule and Lord Kelvin were undertaken for the 
purpose of estimating the deviation of certain actual gases from 
the ideal perfect gas. The gases tested were forced through a 
porous plug and the temperatures on the two sides of the plug 
were accurately determined. In the case of hydrogen the tem- 
perature after passing through the plug was slightly higher 
than on the high pressure side ; air, nitrogen, oxygen, and car- 
bon dioxide showed a drop of temperature. 

From the general law of throttling, the heat content remains 
constant during the process ; that is. 



276 THROTTLING PROCESSES [chap, xiv 

For an ideal perfect gas, 

u-^^ = Jc^T-{- Wq, and pv = BT; 

hence, {Jc, ^ B)T^ = {Jc, + B) T^ 

or ' T^=T^' 

It follows that a perfect gas would show no change of tempera- 
ture in passing through the plug, and that the change of temper- 
ature observed in the actual gas is, in a way, a measure of the 
degree of imperfection of the gas. The results of the experi- 
ments have been used to reduce the temperature scale of the 
air thermometer to the Kelvin absolute scale. 

The ratio of the observed drop in temperature to the drop in 

pressure, that is, the ratio — — , is called the Joule-Thomson 

coefficient and is denoted by ^l. According to the experiments 
of Joule and Kelvin ^ varies inversely as the square of the 
absolute temperature. That is, 

i^-f,- a) 

It may be assumed that this relation holds good for air, nitro- 
gen, and other so-called permanent gases within the region of 
ordinary observation and experiment. At very low tempera- 
tures it seems probable that /x varies with the pressure as well 
as with the temperature. 

An expression for /a in the case of superheated steam can 
readily be derived from the formula for the heat content, namely : 

i = «r+ I /32 2 _ ^ mCy 1) ^^ ^ pyAcp + V 

Since ^ is constant in a throttling process, we may define the 
Joule-Thomson coefficient more accurately as the derivative 

From calculus, we have 

'dT\ _ _ _^ 
dp Ji ~ di/ 
dT 



ART. 166] EQUATION OF THE PERMANENT GASES 
and from the definition of the heat content z, 



Hence 



or 



277 







_(^T\ __1 di 
\dpji Cpdp' 


A 


["%l'\l + «i»+^] 



(2) 



The following table contains values of /^ calculated from 
Eq. (2). 



Pkessure 
Lh. per 
Sq. In. 


250° F. 


300° 


350° 


400° 


450° 


500° 


550° 


600° 


15 
100 

300 


0.668 


0.492 


0.869 
0.327 


0.282 
0.261 


0.220 

0.208 
0.191 


0.176 
0.169 
0.162 


0.143 
0.140 
0.138 


0.119 
0.118 
0.118 



It will be observed that the value of /^ varies with the pres- 
sure ; however, as the temperature rises, the influence of 
pressure decreases ; hence for gases far removed from the satu- 
ration limit, such as were used in the porous plug experiments, 
it seems probable that /i. is a function of the temperature only, 
as found by Joule and Kelvin. 

Dr. Davis has deduced from the throttling experiments of 
Grindley, Griessmann, Peake, and Dodge values of /i for super- 
heated steam.* These were found by direct measurement of 
the mean slopes of the throttling curves. The values thus 
obtained agree very closely with those calculated from (2) and 
shown in the preceding table. 

166. Characteristic Equation of the Permanent Gases. — From the coohng- 
effect shown in the Joule-Thomson's experiments for all gases except hydrogen, 
it appears that those gases do not follow precisely the law expressed by the 

equation pv = BT. By making use of the relation /x = -^ it is possible to 

derive a characteristic equation that represents more nearly the behavior of 
such gases. 

* Proceedings of the Amer, Acad, of Arts and Sciences, Vol. 45, p. 261, 



278 THROTTLING PROCESSES [chap, xiv 

Since the heat content i is constant during a throttling process, the gen- 
eral equation 

di = CpdT - A It p- ~ v\ dp 

takes the form 

c,^ = a(t^-v]. (1) 

"^ dp \ dT I 

Differentiating both members of (1) with respect to T, we obtain 

dT\ dp J \d7' dT^ dTl 

= AT^^, C2) 

But we have 

dT _ _ a 
~dp-^~Y^' 

and from the general thermodynamic relations, 

AT^ = -[^M . 
dT^ \dp)T 

Substituting these expressions in (2), we obtain 

d [ a\ , (dc. 






whence 

'dcp\ , a fdc„\ 2 ac^ 

'/> 

This is a partial differential equation, the general solution of which is the 
equation 

Cp=T^<f>(T^-dap). (4) 

Here <^ denotes an arbitrary function which must be determined from 
physical considerations. Since at high temperatures Cp for permanent gases 
is given by the linear relations Cp — a -{- bT, we have from (4), 

a+bT= T^cf>(T^), 
w^hence <^(r3) = |^^ + y,, 

and <f>(T^-3ap)^ ^ -+ ^ ^. (5) 

(T^-dap)^ {T^-^apY 

Since the term 3 ap is small in comparison with the term T% we have 
approximately 



{T^-^ap) ~' = T-^{\+^^. 



ART. 166] EQUATION OF THE PERMANENT GASES 279 

Introducing these expressions in (5) and substituting the resulting expres- 
sion for <f>{T^ — Sap) in (4), we obtain finally 



'^.=«(i + ^) + "K^+^,) 



= aOr + ffi(^^ + *). (6) 

It appears from (6) that the specific heat of the permanent gases varies 
with the pressure and temperature. At very high temperatures the term 
containing JO is small and the specific heat is given simply by a + bT; at 
low temperatures, however, this term becomes appreciable and the specific 
heat increases with the pressure. The specific heat curves have, therefore, 
somewhat the form shown in Fig. 71. 

From (6), we have by differentiation 

dT'^ AT^\T I 
Integrating, we obtain 

dv a ('2 a , 1 6 \ , ^ . >, .-v 

•J 

Introducing in (1) the expression for — ^ given by (7), we obtain after 



reduction 

V 

T 



'-=f(p) ^[l^ + h + ^(^ + b)\ (8) 

To determine the function /(i?), we assume that the perfect gas equation 

TO 

pv = BT holds when T is very large. Hence /(j») = — , and (8) becomes 

P 

^ ATUST 2 ^ T^\ T /J ^^ 

Since the last term in the bracket is very small, it may be neglected, and (9) 
may be written 

p,= BT-^(^^ + ^-h]. (10) 

^ AT\ST 2 I ^ ^ 

The equation given by Joule and Thomson, namely, 

p.^BT-^^. (11) 

is equivalent to eq. (10) if the constant b is taken as zero. 

Equation (10) may be taken as the general characteristic equation of 
such gases as air, oxygen, and nitrogen. It is useful in certain investigations 
relating to the liquefaction of gases. 



280 



THROTTLING PROCESSES 



[chap. XIV 



167. Linde's Process for the Liquefaction of Gases. — The 

Joule-Thomson effect has been employed by Linde in a very 
ingenious machine for the liquefaction of gases. A diagram- 
matic sketch of the 
machine is shown in 
Fig. 99. Air (or any 
other gas that is to 
be liquefied) is com- 
pressed to a pressure 
of about 65 atmos- 
pheres and is dis- 
charged into a pipe 
leading to the cham- 
ber e, A current of 
cold water in the 
vessel h cools the air 
during its passage 
from the compressor 
to the receiving cham- 
ber. From c the air 
passes through a valve 
V into a vessel d, in which a pressure of about 22 atmospheres 
is maintained. As a result of the throttling the temperature 
of the air is lowered. Thus, if p^ is the pressure in the chamber 
e and p^ ^^^^ pressure in the vessel d, the drop in temperature is 




Fig. 99. 



f2-h = Y'^(pi-P2)' 



(1) 



The air now passes from vessel d at temperature t^ into the 
space enclosing the chamber c and thence back to the compressor. 
In passing back, the air absorbs heat from the air in <?, and the 
temperature rises from ^3 to the final temperature t^, which is 
nearly the same as the initial temperature t^ Due to this 
cooling, the air in c arrives at the valve v with a temperature t^, 
which becomes lower and lower as the process continues. As 
the temperature t^ sinks the temperature ^3 also sinks, but as 
shown by (1), ^3 sinks more rapidly than t^. Ultimately, the 
value of ^3 drops below the critical temperature of the gas. 



ART. 167] LIQUEFACTION OF GASES 281 

and liquefaction begins. The liquid is collected in the vessel 
d and is drawn off through the discharge pipe c. In the ma- 
chine as actually constructed, the counter-current apparatus, 
which in the diagram is shown as the chamber c and the sur- 
rounding space, consists of two concentric spiral pipes about 
100 meters in length. 

By the use of Linde's machine, liquid air in relatively large 
quantities may be produced continuously by the simple mechan- 
ical process of compression. The necessary lowering of tem- 
perature is effected by repeated throttling of the air, and the 
use of a second refrigerating medium is thereby obviated. 

EXERCISES 

1. steam at a pressure of 120 lb. per square inch with 8 per cent moisture 
is throttled to a pressure of 55 lb. per square inch. Find the quality in the 
final state. 

2. If steam at a pressure of 150 lb. per square inch having a quality of 
0.98 is throttled, at what pressure does it become dry saturated steam ? 

3. Find the increase of entropy in each of the preceding examples. 

4. Find the loss of available energy when the pressure of steam is re- 
duced by wiredrawing from 140 to 130 lb. per square inch. The initial 
quality is 0.95 and back pressure is 5 in. of mercury. 

5. In an experiment with a throttling calorimeter the pressure of the 
steam was 110 lb. per square inch, the pressure in the calorimeter was 16 lb. 
per square inch, and the temperature of the steam in the calorimeter was 
246° F. Required the per cent of moisture in the steam. 

6. For the temperatures given in the table of Art. 165 calculate the 
values of /x for a pressure of 200 lb. per square inch. Take values of Cp from 



the curves of Fig. 72. 

7. From Eq. (6), Art. 166, show that the specific heat curve for constant 
pressure passes through a minimum at some 'definite temperature. 

8. From the characteristic Eq. (10), Art. 166, derive an expression for 
the work done during isothermal expansion. 

9. Derive an expression for the heat added during isothermal expansion. 

REFERENCES 
Throttling of Steam 

Zeuner : Technical Thermodynamics 2, 293. 
Berry : The Temperature-Entropy Diagram, 102. 
Lorenz: Technische Wiirmelehre, 237. 



282 THROTTLING PROCESSES [chap, xiv 

Griessmann : Zeit. des Verein. deutsch. Ing. 47, 1852, 1880. 1903. 
Grindley : Phil. Trans. 194 A, 1. 1900. 
Peake : Proc. Royal Society 76 A, 185. 1905. 

The Joule-Thomson Effect 

Thomson and Joule : Phil. Trans. 143, 357 (1853) ; 144, 321 (1854) ; 152, 

579 (1862). 
Natanson : Wied. Annallen 31, 502. 1887. 
Preston : Theory of Heat, 699. 
Bryan : Thermodynamics, 128. 
Lorenz : Technische Warraelehre, 273. 
Davis : Proc. Amer. Acad. 45, 243. 

Characteristic Equation of Gases 

Zeuner : Technical Thermodynamics 2, 313. 
Lorenz : Technical Warmelehre, 296. 
Plank : Physikalische Zeit. 11, 633. 
Bryan : Thermodynamics, 138. 



CHAPTER XV 



TECHNICAL APPLICATIONS, VAPOR MEDIA 



The Steam Engine 

168. The Carnot Cycle for Saturated Vapors. — Since the 
constant pressure line of a saturated vapor is also an iso- 
thermal, three of the processes of the Carnot cycle are ap- 
proximately attainable in a vapor motor, namely : isothermal 
expansion, adiabatic expansion, and isothermal compression. 
The adiabatic compression might also be accomplished by a 
proper arrangement of the organs of the motor, though in 
practice this is never attempted. Hence, the Carnot cycle is 



«k T, BX^" 


/ 




\ 


/ ^ 


T, 


'A 



^1 Bi 

Fig. 100. 




Fig. 101. 



discussed in connection with vapor motors merely for the pur- 
pose of furnishing an ideal standard by which to compare the 
cycles actually used. 

The Carnot cycle on the TO'-plane and joF^planes, respec- 
tively, is shown in Fig. 100 and 101. The isothermal expan- 
sion AB occurs in the boiler, the adiabatic expansion BO in the 
engine cylinder, the isothermal compression CD in the con- 
denser. To effect the adiabatic compression DA, the mixture 
of liquid and vapor in the state D would have to be compressed 

283 



(5) 



284 TECHNICAL APPLICATIONS [chap, xv 

adiabatically in a separate cylinder and delivered to the boiler 
in the state represented by point A. 

The heat received from the boiler per unit weight of fluid is 

qi = ^1(^6 — ^a) ; (area A^ABB^) (1) 

that rejected to the condenser is 

^2 = r^Cx, - xa) ; (area B^ ODA^ (2) 

and the heat transformed into work, represented by the cycle 
area, is 

AW=q^-q^ = ^i^ r^ix, - x^~). (3) 

The efficiency is 

T — T 

and the weight of fluid used per horsepower-hour is 
^^ 2546 2546 T. 

9l - % *'l(^6 - ^a) ^1 - ^2 

If point A lies on the liquid line s' and point B on the satu- 
ration curve s", then x^ =0, rr^, = 1, and (3) and (5) become, 
respectively, 

AW=q,-q, = r,^X^, (3') 

2546 T 

Example. Let the upper and lower pressures be respectively 125 lb. 
per square inch absolute, and 4 in. of mercury. Then from the steam table 
Tx = 804, T2 = 585.1, ri = 875.8 B.t.u. From (4), the efficiency is 

804 - 585.1 ^ . ^„^ 
804 

The heat transformed into work is 875.8x0.272 = 238.2 B.t.u., and the 

2546 
steam consumption is — ; — - — 10.7 lb. per h. p. -hour. 

169. The Rankine Cycle. — In the actual engine the iso- 
thermal compression is continued until the vapor is entirely 
condensed, thus locating the point I) on the liquid curve s'. 
Fig. 102. The liquid is then forced into the boiler by a pump 
and is there heated to the boiling temperature t^. This heat- 



ART. 169] 



THE RANKINE CYCLE 



285 



ing is represented by the segment DA of the liquid curve. If 
it be assumed that the cylinder has no clearance, the pV- 
diagram necessarily takes the form shown in Fig. 103. 




o D, Ar 



Fig. 102. 



Fig. 103. 



The heat supplied from the boiler per pound of steam is in 
this case 

<lx = (Ida + qab = qi - q^l + r^x^ ; 

and the heat rejected to the condenser is 



92 — ^2'^c' 

Hence, the heat transformed into work is 

9i-92 = 9i - 92 + Vb - Vc^ 
and the efficiency of the cycle is 

9i -92 - '^-^ 

9i 



V = 



1- 



(1) 
(2) 
(B) 

(4) 



9i - 92 + Vh 

It is evident that this efficiency is less than that of the Carnot 
cycle. 

The steam consumption per h. p. -hour is 

2546 2546 



N= 



9l - 92 9\ - 92 + ^l^& - ^2^0 



The quality at point C is determined from the relation 

Sl + y^ - «2 + y^ • 



(5) 



(6) 



Example. — Using the data of the example of the last article, determine 
the efficiency and steam consumption of an engine running in a Rankine 
cycle with dry steam. 



286 



TECHNICAL APPLICATIONS 



[chap. XV 



From the steam table, ri = 875.8, r-2 



0.4957, 6-2' = 0.1739, 



ri 



1.0893, ^ 



1023.7, 5i' = 315.2, q.J = 93.4, 
1.7497. The quality at the 



end of adiabatic expansion is 

0.4957 - 0.1739 + 1.0893 



0.806. 



1.7497 
The available heat is 

315.2 - 93.4 + 875.8 - 0.806 x 1023.7 = 272.4 B. t. u. j 

while the heat supplied in the boiler is 

315.2 - 93.4 + 875.8 = 1097.6 B. t. u. 
Hence the efficiency is 

OTO A 

0.248, 



272.4 



-n 



1097.( 



which may be compared with the efficiency 0.272 of the Carnot cycle under 
similar conditions. 

The steam consumption is 

N = ^^ = 9.35 lb. per h. p.-hour. 
272.4 

170. Rankine's Cycle with Superheated Steam. — If super- 
heated steam is used, the Rankine cycle has the form shown in 

Fig. 104. The heat q-^ supplied 
from the source is increased by 
the heat represented by the area 
B^BEE^^ which comes from the 
superheater; and the heat avail- 
able for transformation into work 
is increased by the amount repre- 
sented by the area FBEC. Evi- 
dently the efficiency of the ideal 
cycle is increased by the use of 
superheated steam, but the in- 
crease is small. The advantage of 
superheated steam lies in another 
direction. 
If T^ denote the temperature of the superheated steam (i.e. 
at point E'), the heat required for the superheating process BE 

is J Cj,dT whQVQ-ep is the specific heat of superheated steam. 




Fig. 104. 



AET. 170] CYCLE WITH SUPERHEATED STEAM 287 

This integral may be evaluated by introducing the expression 
for Cp given by Eq. (9), Art. 133. Then the heat represented 
by the area D^DABJSU^ is given by the expression 



^1 



= 91^-92' + r,+ C'c,dT. (1) 



However, as has been shown, the sum q-^' + r^ + \ c^dT is 

practically equal to the heat content of the steam in the state JE. 
Hence we may write 

qi = h-%' (2) 

and calculate % from the general formula (5) Art. 135. 

If the point (7 at the end of adiabatic expansion lies in the 
saturated region, as is usually the case, we have, as in the first 

case, ^2 = Vc 

The heat transformed into work is, therefore, 

qi-% = i-92 -Vc^ (^) 



and the efficiency is 



^e-q2 



The value of x^ is determined from the relation 

«. = V + ^. (5) 

where s^ is the entropy for the state ^, and is calculated from 
the general equation (3), Art. 137. 

Example. — Find the efficiency of the Rankine cycle, using the data of 
the previous examples, but assuming the steam to he superheated to 1000° 
absolute. 

From (6), Art. 185, the heat content of the superheated steam is 

i = 1000(0.367 + 0.00005 x 1000) - 125(1 + 0.0003 x 125) — ^^ -0.0163 x 125 

+ 886.7 = 1294.8 B.t.u.; 
and from (4), Art. 137, the entropy is 

s = 0.8451 log 1000 + 0.0001 x 1000 - 0.2542 log 125 

- 125(1 + 0.0003 X 125) -^ - 0.3964 = 1.7002. 



288 



TECHNICAL APPLICATIONS 



[chap. XV 



Hence 



1.7002 - 0.1739 



= 0.872. 



1.7497 

Heat supplied = i - q^' = 1294.8 - 93.4 = 1201.4 B. t. u. 
Heat rejected = r^Xc = 1023.7 x 0.872 = 892.7 B. t. u. 
Available heat = 1201.4 - 892.7 = 308.7 B. t. u. 

308.7 



Efficiency 



1201.4 



0.257. 



Steam consumption = 



2546 

308.7 



= 8.25 lb. per h. p.-hour. 



171. Incomplete Expansion. — Because of the very large 
specific volume of saturated steam at low pressures, it is usu- 
ally impracticable to continue the adiabatic expansion down 
to the lower pressure p^. The exhaust valve opens and re- 
leases the steam at a pressure somewhat higher than p^. The 
passage of the steam from the cylinder is an irreversible pro- 
cess in the nature of a free expansion and is indicated on the 
pF-diagram by the drop in pressure UF (Fig. 106). The 




Fig. 105. 




Fig. 106. 



actual process may be replaced by an assumed reversible pro- 
cess, cooling at constant volume. On the ^^AS'-diagram the 
cooling is represented by a constant volume line UF (Fig. 
105) drawn as described in Art. 125. 

Evidently this " cutting off the toe " of the diagram results 
in a decrease in the ideal efficiency, but it is justified by the 
smaller cylinder volume required (^DF instead of i>(7) and by 
other considerations. 

Denoting by p^ the pressure at FJ, the end of adiabatic 
expansion, we have: 



ART. 172] CHANGING THE LIMITING PRESSURES 



289 



— /v ' 



q^ + ^6^1- 



Heat absorbed by medium 

^1 = q\ 
Heat rejected by medium 

% = qef + qfd 

Heat transformed into work 

^1-^2= qi-^Vi-Cqs-^^ePs) -^fCr^-P2)' 
The qualities x^ and :cy are found from the equations 



(?2' 



^/Pi) + ^iW 



(1) 



(2) 



(S) 



?l'+^=?3'+^> 



and 



(4) 
(5) 



If the steam is admitted throughout the stroke without cut- 
off, the adiabatic expansion is lacking, and the diagram takes 
the form ABai> (Figs. 105 and 106). The equations for this 
case are readily derived from the preceding equations by 

172. Effect of changing the Limiting Pressures. — If the 
upper pressure p^ be raised to p-^ while the lower pressure p^ 
is kept the same, the effect is to 
increase both q^^ the heat absorbed, 
and q^ — q^^ the available heat, by 
an amount represented by the area 
AAIBB (Fig. 107). Evidently 
the ideal efficiency is thus in- 
creased. If ^2 be lowered to p^^ 
keeping p^ the same, q^ is decreased 
and q^ — q^ increased without any 
change in q^ For the ideal 
Rankine cycle the increase of avail- 
able heat would be that represented 
by the area B' DCC . For the 

modified cycle with incomplete expansion, however, the in- 
crease is represented by the relatively small area B' DFF', 




Fig. 107. 



290 TECHNICAL APPLICATIONS [chap, xv 

We may draw the conclusion that in the actual steam engine 
the limitation imposed by the cylinder volume prevents us 
from realizing much improvement in efficiency by lowering the 
back pressure p^. Herein lies one important difference be- 
tween the steam engine and steam turbine. With the turbine, 
as will be shown, a lowering of the condenser pressure results 
in a marked increase of efficiency. 

173. Imperfections of the Actual Cycle. — In the discussion of 
the ideal Rankine cycle the following conditions are assumed : 

1. That the wall of the cylinder and piston are non-conduct- 
ing, so that the expansion after cut-off is truly adiabatic. 

2. Instantaneous action of valves and ample port area so 
that free expansion or wiredrawing of the steam may not occur. 

3. No clearance. 

In the actual engine none of these conditions is fulfilled. The 
metal of the cylinder and piston conducts heat and there is, 
consequently, a more or less active interchange of heat, between 
metal and working fluid, thus making adiabatic expansion im- 
possible. The cylinder must have clearance, and the effect of 
the cushion steam has to be considered. The valves do not act 
instantly and a certain amount of wiredrawing is inevitable. 
It follows that the cycle of the actual engine deviates in many 
ways from the ideal Rankine cycle, and that the actual efficiency 
must be considerably less than the ideal efficiency. We must 
regard the Rankine cycle as an ideal standard unattainable in 
practice but approximated to more and more closely as the im- 
perfections here noted are gradually eliminated or reduced in 
magnitude. 

The effects of some of these imperfections may be shown 
quite clearly by diagrams on the T/S'-plane. 

In Fig. 108 is shown the cycle of a non-condensing steam 
engine. The feed water enters the boiler in the state represented 
by point G- and is changed into dry saturated steam at boiler 
pressure, represented by point B. When this dry steam is 
transferred to the engine cylinder, which has been cooled to 
the temperature of the exhaust steam, it is partly condensed, 
and the state of the mixture in the cylinder at cut-off' is repre- 



ART. 174] 



EFFICIENCY STANDARDS 



291 



sented by point G. The heat thus absorbed by the cylinder 
walls is represented by the area B^BCC^. CD represents the 
adiabatic expansion, i>£^ the assumed constant-volume cooling 
of the steam, and EF the condensation of the steam at the tem- 
perature corresponding to the back pressure, which is slightly 
above atmospheric pressure. To close the cycle, the water at 
the temperature represented by F 
(somewhat above 212°) must be 
cooled to the original tempera- 
ture of the feed water ; this 
process is represented by FG. 

The heat supplied is repre- 
sented by the area G-^G-ABB^^ 
the heat transformed into work 
by the area FACDE. It will 
be observed that two segments 
of the cycle, namely, GF and 
CB^ are traversed twice, and the 
effect is a serious loss of effi- 




O G,H^ 



A 



Fig. 108. 

IS a serious loss of effi- 
ciency. The loss due to starting the cycle at point G instead 
of at point F may be obviated to a large extent by the use of a 
feed water heater. The heat rejected in the exhaust is used to 
heat the feed water to a temperature represented by point jff", 
which is only a little lower than the temperature of the ex- 
haust. The area G^GHH^ represents the saving in the heat 
that must be supplied. The loss due to cylinder condensation, 
which is shown by the segment BO^ cannot be wholly obviated; 
it may be reduced, however, by superheating and jacketing. 

Losses due to wiredrawing and clearance are not shown on 
the diagram. The drop of pressure in the steam main and in 
the ports may be taken into account roughly by drawing a line 
A^ C somewhat below the line AB^ which represents full boiler 
pressure. 



174. Efficiency Standards. The ratio of the heat transformed 
into useful work to the total heat supplied is usually termed the 
thermal efficiency of the engine. The thermal efficiency, how- 
ever, does not give a useful criterion of the good or bad qualities 



292 TECHNICAL APPLICATIONS [chap, xv 

of an engine for the reason that it does not take account of the 
conditions under which the engine works. It has become cus- 
tomary, therefore, in estimating engine performance to make 
use of certain other ratios. 

Let q = heat supplied to the engine per pound of steam, 

^^= heat transformed into work by an engine working 

in an ideal Rankine cycle (Art. 169), 
q^ = heat transformed into work by actual engine under 

the same conditions, 
Wa = work equivalent of heat q^^ the indicated work, 
Wb = the work obtained at the brake. 
We have then 

77^ = — = thermal efficiency of ideal Rankine engine, 

Q 
7]^ = — = thermal efficiency of actual engine, 

V Q 

T/i == — = — = efficiency ratio (based on indicated work), 

AW 

^ — ~ = brake efficiency ratio (based on work at 

brake), 

W 

T}^ = —^ = mechanical efficiency. 

The ratios rji and t;^ are sometimes called the potential efficiencies 
of the engine, the first the indicated potential efficiency, the 
second the brake potential efficiency. When the term efficiency 
is used without qualification it usually means the efficiency ratio 
or potential efficiency rather than the thermal efficiency. 

It is clear that the useful criterion of the performance of an 
engine is the ratio %. We have 

Vb = Vi X Vm- 

Of the heat q supplied, only the heat g^ could be trans- 
formed into work by the ideal engine using the Rankine cycle ; 
hence the heat q^^ rather than the total heat q should be charged 

Q 

to the engine. The ratio 77^ = — is a measure of the extent to 



ART. 174] EFFICIENCY STANDARDS 293 

which the engine transforms into work the heat ^^ that may 
possibly be thus transformed ; it may be called the cylinder 
efficiency. The ratio rj^ measures the mechanical perfection of 
the engine. Hence, the product rji x rj^ measures the perform- 
ance of the engine both from the thermodynamic and the 
mechanical standpoints. 

The efficiencies rji and rji, may be given other equivalent defi- 
nitions that are frequently useful. 

Let iV^ = steam consumption of ideal Rankine engine per 
h. p.-hour. 
iV^ = steam consumption per h. p.-hour of actual engine. 
N't, = steam consumption per b. h. p.-hour of actual 
engine. 

Then .. = |, .. = |. 

Example. An actual engine operating under the conditions defined in 
the example of Art. 169 shows a steam consumption of 14.1 lb. per i. h. p.- 
hour and 18 lb. per b. h. p.-hour. Since for the ideal engine the steam 
consumption is 9.35 lb. per h. p.-hour, we have 

Vi = ^= 0.663, and r], = ^ = 0.52. 

EXERCISES 

In Ex. 1 to 5 find the heat transformed into work, efficiency, and steam 
consumption per h. p.-hoar. 

1. Carnot cycle, p^ = 110 lb., jOg = 15 lb. absolute, Xf, = 0.85. 

2. Rankine cycle, same data as in Ex. 1 . 

3. Rankine cycle, p^ = 110 lb., P2 = ^ in. of mercury, steam superheated 
to 450" F. 

4. Rankine cycle p^ = 110 lb., J02 = 15 lb., Xi, = 0.85 and adiabatic ex- 
pansion carried to 27 lb. per square inch. 

5. Data the same as Ex. 4 except that steam is not cut oif . 

6. Let P2 be fixed at 5 in. of mercury. Take x^, = 1 and draw a curve 
showing the relation between r) and jOi. Rankine's cycle. 

7. Taking the data of Ex. 2, find the increase of available heat and effi- 
ciency when a condenser is attached and J02 is lowered to 5 in. of mercury. 

8. Make the same calculation for the cycle with incomplete expansion,, 
Ex. 4, and compare the results. 



294 TECHNICAL APPLICATIONS [chap, xv 

. 9. The efficiency rji of an engine is 0.65 and the mechanical efficiency is 
0.85. If the heat transformed into work by the ideal Rankine engine is 
190 B. t. u. per pound, what is the steam consumption of the actual engine 
per b. h. p.-hour? 

10. The steam consumption of a Rankine engine is 9.2 lb. per h. p.- 
hour, and the efficiency ratio rji is 0.70. Find the heat transformed into 
work by the actual engine per pound of steam. 



The Steam Turbine 

175. Comparison of the Steam Turbine and Reciprocating En- 
gine. — The essential distinction between the two types of 
vapor motors — turbines and reciprocating engines — lies in 
the method of utilizing the available energy of the working 
fluid. In the reciprocating engine this energy is at once util- 
ized in doing work on a moving piston ; in the turbine there is 
an intermediate transformation, the available energy being first 
transformed into the energy of a moving jet or stream, which is 
then utilized in producing motion in the rotating element of the 
motor. 

While the turbine suffers from the disadvantage of an added 
energy transformation with its accompanying loss of efficiency, 
it has a compensating advantage mechanically. With any 
motor the work must finally appear in the rotation of a shaft. 
Hence, intermediate mechanism must be employed to transform 
the reciprocating motion of the piston to the rotation required. 
Evidently this is not the case with the turbine, which is thus 
from the point of view of kinematics a much more simple ma- 
chine than the reciprocating engine. Many attempts have been 
made to construct a motor (the so-called rotary engine) in 
which both the intermediate mechanism of the reciprocating en- 
gine and the intermediate energy transformation of the turbine 
should be obviated. These attempts have uniformly resulted 
in failure. 

With ideal conditions it is easily shown that the two methods 
of working produce the same available work and, therefore, 
give the same efficiency with the same initial and final con- 
ditions. Thus the Rankine ideal cycle. Fig. 102, gives the 
maximum available work per pound of steam of a reciprocating 



ART. 176] CLASSIFICATION OF STEAM TURBINES 295 

engine with the pressures p^ and p^. It Hkewise gives (Art. 
152) the kinetic energy per pound of steam of a jet flowing 
without friction from a region in which the pressure is p^ into 

2 

a region in which it is p^. Hence if this kinetic energy — 

2g 

is wholly transformed into work, the work of the turbine per 
unit weight of fluid is precisely equal to that of the reciprocat- 
ing engine. Under ideal conditions, therefore, neither type of 
motor has an advantage over the other in point of efficiency. 

Under actual conditions, however, there may be a consider- 
able difference between the efficiencies of the two types. Each 
type has imperfections and losses peculiar to itself. The re- 
ciprocating engine has large losses from cylinder condensation ; 
the turbine, from friction between the moving fluid and the 
passages through which it flows. It is a question which set of 
losses may be most reduced by careful design. 

Aside from the question of economy, the turbine has certain 
advantages over the reciprocating engine in the matters of 
weight, cost, and durability (associated with certain disadvan- 
tages) and these have been sufficient to cause the use of tur- 
bines rather than reciprocating engines in many new power 
plants and also in some of the recently built steamships. 

176. Classification of Steam Turbines. — Steam turbines may 
be divided broadly into two classes in some degree analogous 
to the impulse water wheel and the water tur- 
bine, respectively. In the first class, of which 
the de Laval turbine may be taken as typical, 
steam expands in a nozzle until the pressure 
reaches the pressure of the region in which the 
turbine wheel rotates. The jet issuing from 
tlie nozzle is then directed against the buckets 
of the turbine wheel. Fig. 109, and the impulse 
of the let produces rotation. It will be noted 

Fig. 109. 

that with this type of turbine only a part of the 

buckets are filled with steam at any instant, even if several 

nozzles are used. 

In turbines of the second class, the steam flows through guide 





296 TECHNICAL APPLICATIONS [chap, xv 

vanes in a stationary ring s, Fig. 110, and then through blades 
in the circumference of the moving wheel m. The guides and 
wheels " run full," that is, the stationary and moving blades 
are filled with steam throughout the entire circumference. 
The pressure of the steam is reduced during the 
passage through the blades both in the guide and 
turbine wheels. In the turbine of the first type all 
the available internal energy of the steam is trans- 
formed into kinetic energy of motion before the 
steam enters the turbine wheel, while in the turbine 
of the second type part of the internal energy is 
transformed into work during the passage of the 
fluid through the wheel. 

The terms impulse and reaction have been used 
Fig. 110. to designate turbines of the first and second class, 
respectively. Since, however, impulse and reaction 
are both present in each type, these terms are somewhat mis- 
leading, and the more suitable terms velocity and pressure have 
been proposed. Thus a de Laval turbine is a velocity turbine ; 
a Parsons turbine is a pressure turbine. 

177. Compounding. — The high velocity of a steam jet result- 
ing from a considerable drop of pressure renders necessary 
some method of compounding in order that the peripheral 
speed of the turbine wheels may be kept within reasonable 
limits without reducing the efliciency of the turbine. With 
velocity turbines three methods of compounding are employed. 

1. J^ressure Compounding. The total drop of pressure ^^ —jt?2 
may be divided among several wheels, thus reducing the jet 
velocity at each wheel. If, for example, the change of heat 
content is ii — i^ and the expansion takes place in a single 
nozzle, the ideal velocity of the jet is w = V^gJi^i-^^ — i^) ; 
if, however, i^ — i^ is divided equally among n wheels, the jet 

velocity is reduced tow— \-^— (i^ — i^. The general arrange- 

n 

ment of a turbine with several pressure stages is shown in 

Fig. 111. Steam passes successively through orifices m-^^ m^^ 

etc. in partitions h^, h^, etc., which divide the interior of the 



ART. 177] 



COMPOUNDING 



297 




Fig. 111. 



turbine into wheel chambers. The pressure drops from p^ to 
P2 in the first cell and the jet acts on the first wheel ; then in 
passing through the orifice Wg the pres- 
sure drops from p^ to p^; as a result 
the velocity is again increased and the 
jet passes through the second wheel. 
The pressure and velocity changes are 
shown roughly in the diagram at the 
bottom of the figure. 

The method of compounding here 
described is called pressure compound- 
ing. Each drop in pressure constitutes 
a pressure stage. 

2. Velocity/ Compounding. The steam 
may be expanded in a single stage to 
the back pressure p^, thus giving a rela- 
tively high velocity ; and the jet may 
then be made to pass through a suc- 
cession of moving wheels alternating 

with fixed guides. This system is shown diagrammatically in 
Fig. 112. The jet passes into the first moving wheel, where 
it loses part of its absolute velocity, as indicated by the 
velocity curve w. It then passes through 
the fixed guide g^ with practically con- 
stant velocity and has its direction 
changed so as to be effective on enter- 
ing the second moving wheel. Here 
the velocity is again reduced and the 
decrease of kinetic energy appears as 
work done on the wheel. This process 
may be again repeated, if desired, by 
adding a second guide and a third wheel. 
However, the work obtainable from a 
wheel is small after the second moving 
wheel is passed, and a third wheel is 
not usually employed. 

3. Combination of Pressure and Velocity Compounding. 







^r=^ 



^ 






9i 
Fig. 112. 






Evi- 



dently the two methods of compounding may be combined in a 



298 



TECHNICAL APPLICATIONS 



[chap. XV 



variety of ways. The Curtis turbine, which is a well-known 
representative type, has usually four or five pressure stages 
with two velocity stages to each. That is, there are four or 
five sets of nozzles delivering steam to a corresponding number 
of wheels running in separate chambers, and each wheel has 
two sets of blades separated by guide vanes. 

Pressure turbines are always of the multiple pressure-stage 
type, and the number of stages is large. The arrangement is 

that shown in Fig. 113. 
The steam flows through 
alternate guides and moving 
blades, its pressure falling 
gradually as indicated by 
the curve pp. The absolute 
velocity of flow increases 
through the fixed blades 
and decreases in the moving- 
blades as indicated by the 
velocity curve ww. This 
curve, it will be observed, 
rises as the pressure falls 
much as if the turbine were 
a large diverging nozzle. 
The steam velocity with 
this type of turbine is, however, relatively low even in the 
last stages. 




Fig. 113. 



178. Work of a Jet. — While the problems relating to the 
impulse and reaction of fluid jets belong to hydraulics, it is 
desirable to introduce here a brief discussion of the general case 
of the impulse of a jet on a moving vane. 

Let the curved blade have the velocity e in the direction in- 
dicated. Fig. 114, and let w^ denote the velocity of a jet directed 
against the blade. The velocity w-^^ is resolved into two compo- 
nents, one equal to <?, the velocity of the blade, the other, there- 
fore, the velocity a-^ of the jet relative to the blade. The angle 
of the blade and the velocity c should be so adjusted that the 
direction of a^ is tangent to the edge of the blade at entrance. 



ART. 178] 



WORK OF A JET 



299 



The jet leaves the blade with a relative velocity a^ equal in 
magnitude to a^, neglecting friction, but of less magnitude if 
friction is taken into account. This velocity a^ combined with 
the velocity c of the blade gives the absolute exit velocity w,^: 
It is convenient to draw all the velocities from one point as 
shown in the velocity diagram. 

The absolute entrance and exit velocities w-^ and w^ may be 
resolved into components w^ and w^ in the direction of the 
motion of the vane 



and 



UK 



and 



w. 




Fig. 114. 



at 
right angles to this 
direction, that is, 
parallel to the axis of 
the wheel that carries 
the vane. These 
latter may be termed 
the axial components, 
the former the 'pe- 
ripheral components. 
The driving impulse 
of the jet depends 
upon the change in 

the peripheral component only. To deduce an expression for 
the impulse we proceed as follows ; 

Let Am denote the mass of fluid flowing past a given cross 
section in the time A^ ; then the stream of fluid in contact 
with the blade may be considered as made up of a number of 
mass elements Am, and in the time element A^ one mass ele- 
ment enters the vane with a peripheral velocity w^ and another 
leaves it with a peripheral velocity w^. The effect is the same 
as if a single element Aw by contact with the blade had its 
velocity decreased from w-l to w^ in the time td. From the 
fundamental principle of mechanics, the force required to pro- 



duce the acceleration — ^ 



A^ 



IS 



V 



Am ^V-^i 



A^ 



(1) 



This force is the force exerted by the blade upon the element 



300 



TECHNICAL APPLICATIONS 



CHAP. XV 



Am; an equal and opposite force is, therefore, the impulse of 
Am on the vane. 

If M denotes the weight of steam flowing per second, then 

M 

Am = — A^, and we have for the force exerted by the jet on 

the vane in the direction of the velocity e, 

<). (2) 



9 



Evidently this equation holds equally well when the weight M 
flowing from the nozzle is divided among several moving vanes. 
The product pc of the peripheral force and peripheral veloc- 
ity of vane gives the work per second ; therefore, 



and 



work per second = — - (w^ — w^)^ 



work per pound of fluid = - {w^ — w^). 



(3) 



(4) 



When, as is usually the case, the direction of w^ is opposite 
to that of w^^ the sign of w^ must be considered negative and 
the algebraic difference w^ — w^ in (2), (3), and (4) becomes 
the arithmetic sum w-^' + w^ . 

179. Single-stage Velocity Turbine. — In analyzing the action 
of the single-stage velocity turbine, it is convenient to start 

with an ideal frictionless tur- 
bine and then take up the 
case of the actual turbine. 

Let the jet emerge from 
the nozzle with the velocity 
w^j. Fig. 115, at an angle a 
with the plane of the wheel. Combining w^ with c^ the periph- 
eral velocity of the blade, the velocity a^ of the jet relative 
to the blade is obtained. The angle /3 between the direction 
of «j and the plane of the wheel determines the angle of the 
blade at entrance. If the blade is symmetrical, the exit relative 
velocity a<^ makes the same angle /3 with the plane of the wheel, 
and since the frictionless case is assumed, a^ = a^ Combining 
^2 and c, the result is the absolute exit velocity w^. 




ART. 179] SINGLE-STAGE VELOCITY TURBINE 301 



The energy of the jet with the velocity iv-^ is — ^ per pound 

nn 2 

of medium flowing ; and the jet at exit has the energy — ^ . 

^9 
The work absorbed by the wheel per unit weight of steam in 
this ideal frictionless case is, therefore, 

Tr=^^P^', (1) 



and the ideal efficiency is 



w-^ — w^ 



(2) 



From the triangle OAE^ Fig. 115, we have 

w^ = w^ + (2 cy - 2 w^(2 c) cos a ; (3) 

whence w-^— w^ = ^{u\ c cos a— c^). (4) 

Combining (2) and (4), we get, 

77 = 4— ( cos a— — ). (5) 

Equation (5) shows that the efficiency is greater the smaller 
the angle a ; and that with a given constant angle cc, the effi- 
ciency depends upon the ratio — . It is readily found that rj 

takes its maximum value r)^^^ = cos^ a when the ratio — takes 



W-, 



the value ^ cos a. 



As an example, let a = 20°, whence cos a =0.9397 and cos^a^ 0.883. If 
w = 3600 ft. per second, then to get the maximum efficiency 0.883, the ratio 

— must be i cos a = 0.47, whence c = 0.47 x 3600 = 1692 ft. per second, a 
wi 2 

value too high for safety. If c be given the permissible value 1200 ft. per 

second, we have —=l, and ri = 4 x - (0.9397 - 0.3333) = 0.809. 
w, o 3 

In the actual turbine, friction in the nozzle and blades reduces 
the efficiency considerably below the value given by (5). The 
velocity diagram with friction is shown in Fig. 116. The ideal 
jet velocity w^ is reduced by friction in the nozzle to Wy This 



302 



TECHNICAL APPLICATIONS 



CHAP. XV 



actual jet velocity w^ combined with velocity e gives the relative 
velocity «j, as before. The exit relative velocity a^ is smaller 

than a^ because of friction 
in the blades, and as a 
result the absolute exit ve- 
locity w^ is smaller than in 
the ideal case. 

The work per pound of 
steam may be found from 
the velocity diagram either by calculation or by direct measure- 
ment. Having the components w^' and w^^ the work per 
pound of steam is given by the expression 




Fig. 116. 



9 



w^). 



(6) 



This work may be compared with the work obtained from the 
ideal frictionless turbine given by (1) or with the energy of 

the jet per pound of steam, namely, —-. 

^9 

180. Multiple -stage Velocity Turbine. — In the Rateau turbine 
and in others of similar construction, the principle of pressure 
compounding is employed. The turbine consists essentially 
of several de Laval turbines in series, running in separate cham- 
bers. See Fig. 111. The action of this type of turbine is con- 
veniently studied in connection with a MoUier diagram. Fig. 117. 

Let the initial state of the steam entering the turbine 
at the pressure f^ be that indicated by the point A. If f^^ 
is the pressure in the first chamber, a frictionless adiabatic 
expansion from ^^ to 'p^ is represented by AB^ and the 
decrease in the heat content ^■^ — i^ is represented by the 
length of the segment AB. Under ideal conditions, this 
drop ill the heat content would all be transformed into 
kinetic energy of the jet of steam flowing into the chamber, 
and this in turn would be given up to the wheel. Actu- 
ally, however, friction losses are encountered and the jet 
has an exit velocity w^^ thereby carrying away the kinetic 

energy ^. The velocity diagram for the single wheel under 



ART. 180] MULTIPLE-STAGE VELOCITY TURBINE 



303 



consideration is similar to that shown in Fig. 116. The work 
lost in overcoming friction in the nozzles and blades and the 



w^ 



exit energy ^ are transformed into heat, and this heat, except 

a small fraction that is radiated, is expended in further super- 
heating (or raising the quality of) the steam. Hence, instead 
of the final state B^ we have a final state Q on the same con- 
stant-pressure curve. Referring to Fig. 117, AO represents 




Fig. 117. 

the part of the heat drop that is utilized by the wheel, while 
C^B represents the part that is rendered unavailable by internal 
losses of various kinds. 

The steam in the state Q flows into the second chamber where 
the pressure is f^. Frictionless adiabatic expansion would give 
the second state i), but the actual state is represented by the 
point H. Again CW represents the effective drop of heat con- 
tent in this stage, while B^J) represents the part of the drop 
going back into the steam. 

The same process is repeated in succeeding stages until 
finally the steam drops to condenser pressure in the last stage. 
The final state is represented by the point K^ and the curve 
J.jE^^ represents the change of state of the steam during its 
passage through the turbine. The final state under ideal fric- 
tionless conditions is represented by point M. The segment 
AM represents the ideal heat drop, which, as has been shown, is 
equal to the available heat of the Rankine cycle. The segment 



304 



TECHNICAL APPLICATIONS 



[chap. XV 



AN represents the heat drop utilized. The ratio 



AJSr 
AM 



de- 



pends upon the magnitude of the internal losses, such as friction 
in nozzles and blades, leakage from stage to stage, windage, 
exit velocity, etc. Roughly, this ratio may lie between 0.50 
and 0.80. 




181. Turbine with both Pressure and Velocity Stages. — In 

certain turbines, notably the Curtis turbine, velocity compound- 
ing is employed. There are relatively few (three to seven) 

pressure stages, 
but in each cham- 
ber there are two 
or three rows of 
moving blades at- 
tached to the 
YiG. 118. wheel rim and 

these are sepa- 
rated by alternate rows of guide blades, as shown in Fig. 112. 

The velocity diagram for a single pressure stage with two 
velocity stages is shown in Fig. 118. The velocities in relation 
to the successive sets of blades are shown in Fig. 119. The jet 
emerges from the nozzle with an absolute velocity w^, which is 
smaller than the ideal w^. 



because of friction in 

nozzle. Combining w^ with 

the peripheral velocity c of 

the first moving blade m^, the 

result is the velocity a^ of 

jet relative to blade m^. The 

angle a between «j and the 

plane of rotation is the proper 

entrance angle of the blade. 

my The exit relative velocity a^^ which is smaller than a-^^ 

due to friction in the blade, is combined with the velocity ^, 

giving the absolute exit velocity w^ which makes the angle /3 

with the plane of rotation. The jet enters the stationary 

guide blade s with the velocity w^ and emerges with a smaller 



'0 

the 




Fig. 119. 



ART. 182] PRESSURE TURBINE ^ 305 

absolute velocity w^. It then enters the second moving 
blade 7722* Combination of w^ with c gives the relative 
velocity ag ^^^^ ^^® entrance angle 7 for the blade Wg. The 
exit velocity a^ is determined from a^ and the friction in the 
blade, and by combining a^ and c, the absolute exit velocity w^ 
is obtained. 

In the diagram, Fig. 119, the blades have been taken as 
symmetrical. Sometimes, however, the exit angles of the last 
sets of blades are made smaller than the entrance angles. The 
diagram can easily be modified to suit this condition. 

The work per pound of steam for this wheel is readily deter- 
mined from the velocity diagram. From the first set of blades 

m^ the work - (^w-^' — w^} and from the second set of blades m^ 

the work - (wj — wJ) is obtained. Hence the total work per 
pound of steam is 

W= - (w^^ - w^ + w^ - iv^). (1) 

Care must be taken that w^ and w^ be given their proper alge- 
braic signs. 

The state of the fluid as it passes through the turbine may 
be shown by the Mollier diagram precisely similar to that 
shown in Fig. 117. Starting with an initial state indicated by 
point A^ the available drop from the initial pressure p^ to the 
pressure p^ in the first chamber is represented by AB. The 
heat utilized in useful work TTas given by (1) is represented 
by AO^. Hence projecting C horizontally to (7 on the line of 
constant pressure p2-> we get the state of the steam as it enters 
the second stage nozzles. 

182. Pressure Turbine. — In the pressure type of turbine 
there is always a large number of stages, the guide blades and 
moving blades alternating in close succession. The fact that 
the pressure falls continuously, both through the guide blades 
and the moving blades, makes the velocity diagram essentially 
different from that of the velocity turbine. Referring to Fig. 1 20, 
let w-^ denote the absolute velocity of the steam entering the 



306 



TECHNICAL APPLICATIONS 



CHAP. XV 



stationary blade Sj, and W2 the absolute exit velocity. If there 
were no change of pressure, W2 would be smaller than w-^ be- 
cause of friction ; but the drop in pressure Ap causes a decrease 
in heat content Ai, and as a result, there 
is an increase of velocity given by the 
relation 



^ 



w„ 



w. 



w 



'^0 



= -VJ(1 - ^)A? 



ai 



w^ 



Fig. 120. 



Thus the exit velocity w^ is greater than 
the entrance velocity w-^^. Combining w^ 
with c, the velocity of the moving blade, 
we obtain a^, the velocity of entrance 
relative to the moving blade. Now the 
pressure drops through the moving 

blades also ; hence as a result the velocity of exit a^ is greater 

than a^, just as Wg' ^^ greater than w^ Combining a^ with <?, 

the result is w-^^ the absolute velocity of entrance into the 

next row of fixed blades. 

The work done in any single stage, consisting of one set of 

stationary blades and one set of moving blades, is obtained from 

the velocity diagram for that stage in the usual way. Thus, 

if we have the diagram shown in 

Fig. 120 for a particular stage, 

the work per pound of steam 

for that stage is given by the 

product 

if 

If the fixed and moving blades 
have the same entrance angles 
and exit angles, it may be as- 
sumed that the velocity diagram 

has the symmetrical form shown in Fig. 121 ; that is, w-^ — a^ 
and w^ = ^2- ^"^ ^^^^ case, the work may be obtained by a simple 
graphical construction. Using point ^ as a center and with a 
radius BA let a circular arc ADOhQ described and from jE71et 
a perpendicular be dropped cutting this arc in I). Denoting the 
length ED by A, we have 




ART. 183] INFLUENCE OF HIGH VACUUM 307 

It follows that the work per pound of steam is given by the 

expression — provided h is measured to the same scale as the 

9 
velocity vectors iv^^ w^. 

183. Influence of High Vacuum. — In Art. 172 it was pointed 
out that the reciprocating engine is unable to take advantage 
of a very low back pressure for the reason that the cylinder 
volume cannot be made sufficiently large to permit the expan- 
sion of the stearti to the condenser pressure. No such restriction 
applies to the steam turbine. The blades in the final stages 
may be made long enough to pass the required volume of steam 
at the lowest pressures obtainable. The advantage of the tur- 
bine in this respect is shown graphically in Fig. 107. Since 
the cylinder volume of the reciprocating engine is limited to 
the volume indicated by the point E^ the effect of lowering the 
back pressure from p^ to p^ is the addition of the area B'DFF' 
to the area of the original cycle. The turbine, however, can 
accomodate volumes indicated by points C and 0' ; hence if the 
pressure is lowered from p^ to p^^ the area of the ideal cycle is 
increased by the area B'BCO'. It is evident, therefore, that 
high vacuum is much more effective in the case of the steam 
turbine than in the case of the reciprocating engine. 

The superior efficiency of the steam turbine at low pressures 
and the ability of the turbine to make effective use of high 
vacuum has led to the introduction of the low-pressure turbine 
in combination with the reciprocating engine. The engine 
takes steam at boiler pressure and exhausts into the turbine at 
about atmospheric pressure. In general, the combination is 
more efficient than either the engine alone or the turbine alone 
using the entire range of pressure. 

EXERCISES 

1. In a single-stage velocity turbine the jet emerges from the nozzle with 
a velocity of 3150 ft. per second and the direction of the jet makes an angle 
of 22^° with the plane of rotation, (a) Find the circumferential velocity 



308 



TECHNICAL APPLICATIONS 



[chap. XV 



that will give maximum efficiency, (b) Find the efficiency if the circum- 
ferential velocity is 1100 ft. per second. 

2. Find the work per pound of steam in case (&) of Ex. 1. 

3. Using the data of Ex. 1 and 2 assume that the exit relative velocity is 
reduced 10 per cent by friction in the blades. Draw a velocity diagram and 
by measurement or calculation find the work done per pound of steam. 
Compare this result with that found for the ideal frictionless case. 

4. A reciprocating engine receives steam at a pressure of 160 lb. per 
square inch, superheated 120^^. The steam expands adiabatically to a pres- 
sure of 16 in. of mercury and is then discharged into a low pressure turbine 
where it expands adiabatically to a pressure of 2 in. of mercury. Find the 
percentage by which the efficiency is increased by the addition of the tur- 
bine. Assume ideal conditions. 

5. A turbine of the Curtis type has three pressure stages. The initial 
pressure is 140 lb. with the steam superheated 120° F., and the condenser 
pressure is 3 in. of mercury. The loss of energy due to friction, etc., is 30 
per cent of the total available energy, (a) Find the condition of the steam 
entering the condenser, (b) Find the consumption per h. p.-hour. (c) 
Determine the intermediate pressures in the cells on the assumption that the 
work developed in each stage shall be approximately the same. 



184. 

organs 



Refigeration with Yapor Media 

Compression Refrigerating Machines. — The essential 
of a compression machine using vapor as a medium 

are shown in Fig. 



-ff3:t-=^ 



Brine Tank | 




[ Condensei 



122. The action of 
the machine may be 
studied to advan- 
tage in connection 
with the TS-dm~ 
gram, Fig. 123. 
The medium is 
drawn into the 
compressor cylinder 
through the suction 
pipe from the coils 
in the brine tank. 
It may be assumed that the medium entering is in the saturated 
state at the temperature T^ which may be taken equal to the 
temperature of the brine. This state is represented by point B, 



Expansion 
Valve 



Fig. 122. 



ART. 184] COMPRESSION REFRIGERATING MACHINES 309 



T 




1 T, \ 


C 

i\ 




r 


L 


A T, 


\ 


p 


H 


/ 




M 




< 



O Gi E^A 



Dr C, 



Fig. 123. The vapor is compressed adiabatically to a final 
pressure jOg, which is determined by the upper temperature T^ 
that may be obtained with the cooling water available. The 
adiabatic compression is represented by BO. The superheated 
vapor in the state O is discharged into the coils of the cooler 
or condenser, where heat is abstracted from it. The coils are 
surrounded by cold water which 
flows continuously. First the 
gas is cooled to the state of 
saturation ; this process is rep- 
resented by the curve (7i>, and 
the heat abstracted by the area 
CjCjDjDj. Then heat is further 
removed at the constant tem- 
perature ^2 (and pressure p^) 
and the vapor condenses. At 
the end of the process, the 
medium is liquid and its state 
is represented by the point U 
on the liquid curve. 

It should be noted that there are two parts of the fluid circuit : 
one including the discharge pipe and coils at the higher pres- 
sure jt?2' 3-nd one including the brine coils and the suction pipe 
at the lower pressure p^. These are separated by a valve called 
the expansion valve. The liquid in the state represented by 
point ^ is allowed to trickle through the valve into the region 
of lower pressure. The result of this irreversible free expan- 
sion is to bring the medium to a new state represented by point 
A, In this state the medium, which is chiefly liquid with a small 
percentage of vapor, passes into the coils in the brine tank or 
in the room to be cooled. The temperature of the brine being 
higher than that of the medium, heat is absorbed by the medium, 
and the liquid vaporizes at constant pressure. This process is 
represented by the line AB and the heat absorbed from the 
surrounding brine by the area A^ABC^ 

The position of the point A is determined as follows : The 
passage of the liquid through the expansion valve is a case of 
throttling or wiredrawing of the character discussed in Art. 162. 



Fig. 123. 



310 TECHNICAL APPLICATIONS [chap, xv 

Hence, the heat content at A must be equal to the heat content 
at U, that is, 

Graphically, the area OUGrAA^ is equal to the area OHEE^ ; or 
taking away the common area OHGrFE^^ the rectangle E^FAA^ 
is equal to the triangle QEF. (See Art. 162). 

Since the throttling process represented by EA is assumed to 
be adiabatic, the work that must be done on the medium is the 
difference between §i, the heat absorbed, and ^21 the heat rejected 
to the condenser. We have then 

(32 = area C^ODEE^, 

Q^ = area A^ABC^, 

W = area C^CDEE^ - area A^ABQ^ 

= area BODEE^A^AB 

= area BODEaB. 

If the expansion valve be replaced by an expansion cylinder, 
permitting a reversible adiabatic expansion from p^ to jt?^, as in- 
dicated by the line EF^ we have 

(32 = area C^CBEE^, 
q\ = area E^FBC^, 
W= area BQDEFB. 

The effect of using the expansion valve rather than the expansion 
cylinder is thus to decrease the heat removed by the area E^FAAl^ 
and to increase the work done by an equivalent amount. 

185. Vapors used in Refrigeration. — The three vapors that are 
used to any extent as refrigerating media are ammonia, sulphur 
dioxide, and carbon dioxide. Of these, ammonia is used almost 
exclusively in America and largely in Europe. The other two 
are used to a small extent chiefly in Europe. 

The choice of vapor to be used depends chiefly upon two things : 
(1) The suction and discharge pressures that must be employed 
to give proper lower and upper temperatures T^ and T^. The 
lower temperature must be such as to keep the proper temperature 
in the brine or the space to be kept cool, while the upper 
temperature is fixed by the temperature of the cooling water 



NH3 


S02 


CO, 


41.5 


14.75 


385 


124 


47.61 


826 


4.4 


12 


1 



ART. 186] CALCULATION OF A VAPOR MACHINE 311 

available. (2) The volume of the medium required for a given 
amount of refrigeration. This determines the bulk of the 
machine. 

If the upper temperature be taken as 68° F. (^2 = 528) and 
the lower temperature as 14° F., the pressures and the volume 
ratios for the three vapors mentioned are about as follows: 

Suction pressure, lb. per sq. in. 
Discharge pressure, lb. per sq. in. 
Volume, taking that of CO2 as 1 

It appears that carbon dioxide requires for proper working 
very high pressures, so high, in fact, as to be practically prohib- 
itive except in machines of small size. With sulphur dioxide 
the pressures are low, but the necessary volume of medium is 
high, being nearly three times that required by ammonia and 
twelve times that required by carbon dioxide. With ammonia, 
the pressures are reasonable and the volume of medium is not 
excessive; hence from these considerations, ammonia is seen to 
be most advantageous. 

From the point of view of economy, ammonia and sulphur 
dioxide are about equal. Carbon dioxide shows a somewhat 
smaller efficiency than the others under similar conditions be- 
cause, on account of the small latent heat of carbon dioxide, the 
losses due to superheating and the passage through the expan- 
sion valve are a larger per cent of the total effect. 

186. Calculation of a Vapor Machine. — -The following analysis 
applies to the ideal cycle shown in Fig. 123. Denoting by T^ 
the temperature at the end of compression indicated by the 
point (7, the heat that must be removed per minute from the 
superheated vapor to bring it to the saturation state (the heat 
represented by the area (7j ODB^ is 

in which c^ denotes the specific heat of superheated vapor, and 
iff", the weight of the medium required per minute. The heat 
rejected by the vapor during condensation (area D^DEE^) is 
Mr^. Hence the heat rejected by the medium per minute is 

Q^ = Mlr^ + c,(Z-T^^-\. (1) 



312 TECHNICAL APPLICATIONS [chap, xv 

Denoting by x^ the quality of the mixture of liquid and vapor 
in the state represented by point A, we have for the heat ab- 
sorbed by the medium from the brine or cold room (repre- 
sented by the area A^ABC^) 

Q, = Mr,a-^i)' (2) 

But area OHGiAA^ = area OHEE^, that is, 

9.1 + Vi = 92 5 (3) 

whence combining (3) and (2), 

Q, = MCr, - q^' + j/) = M(q," - q,'). (4) 

The work required per minute is, therefore, 

W= Ji <?2 - <?i) = JM[_q^' - ?i" + ".( T. - T,-)}, (5) 
and the net horsepower required to drive the machine is. 

Combining (6) and (4), we have 

778(g,[fe"-g/'+.,(r.-y,)] 

33000(j,"-gi') • '-'^ 

To the horsepower thus calculated should be added perhaps 
10 to 20 per cent to allow for imperfections of the cycle, and to 
the gross horsepower must be added 10 to 20 per cent to allow 
for the friction of the mechanism. 

Assuming the vapor entering the compressor to be dry and 
saturated, as indicated by point B, Fig. 123, the volume of 
vapor entering the compressor per stroke is 

^-^ c«) 

where v-^" is the specific volume of vapor at the pressure p^ 
and iV the number of working strokes per minute. If the 
medium enters the compressor as a mixture of quality x^ as in- 
dicated by point M, then approximately 

' N ^ ^ 

The net cylinder volume as determined by (8) or (9) must 



ART. 186] CALCULATION OF A VAPOR MACHINE 313 

be increased 10 to 25 per cent to allow for clearance and 
various imperfections. 

The weight of cooling water required per minute is readily 
found from (1) when the initial and final temperatures of the 
water are fixed. Denoting this weight by Gr and the initial and 
final temperatures by t'^ and t\ respectively, we have 

a(t"-t')=Mir^ + e^iT,-T^-)-]. (10) 

To determine the value of Q^^ from (1) the temperature T^ at 
the end of compression must be obtained. For adiabatic com- 
pression Tc may be found by the following method. Referring 
to Fig. 123, the decrease of entropy in passing from C to D is 
the same as passing from B to D. If Cp, the specific heat along 
curve CD, is assumed to be constant, we have 





^%- «d= (^pi^og^Y' 


But 


.,-.,=v+|f-(< + ^ 


hence 





(11) 

Since c^, 2\, T^, s^^ Sg, /*!, and rg are known quantities, T^ is 
easily calculated. 

Example. Required the dimensions and the horsepower of an ammonia 
refrigerating machine that is to abstract 15,000 B. t. u. per minute from a 
cold chamber which is to be kept at a temperature of 30° F. The tempera- 
ture of the ammonia in the condenser may be taken as 85° F. and that of 
the ammonia in the brine coils 20° F. Assume one double-acting com- 
pressor making 75 r. p. m. 

From the table of the properties of saturated ammonia, we have the fol- 
lowing values corresponding to tx — 20° and «2 = 85° : 

jwi = 47.46 lb. per square inch, ri = 560 B. t. u., qx = — 13 B. t. u., 

^i" = 547 B. t. u., si' = - 0.027, ^ = 1.168, v^' = 6.01 cu. ft.. 

Pi = 166.8 lb. per square inch, ri = 496 B.t. u., q^' = 61 B.t.u., 

qi" = 557 B.t.u., S2' = 0.118, -^ = 0.910, V2" = 1.78. 

T2 



314 TECHNICAL APPLICATIONS [chap, xv 

To determine the temperature T^ of the superheated ammonia at the 
end of compression, we have, from (11), 

0.51 loge -^ = - 0.027 + 1.168 - (0.118 + 0.910)= 0.113, 
544.6 ^ 

whence log T, = log 544.6 + 0.4343 x ^^ = 2.83231, 

0.51 

T,= 679.7, 

and t, = 679.7 - 459.6 = 220.P F. 

The weight of ammonia that must be circulated per minute is, from (4), 

J.f=^^= JfO» =30.86 lb. 
^/' - q^' 547 - 61 

The net horsepower is, from (6), 

778 X 30 



33000 



[557 - 547 + 0.51(220.1 - 85)]= 57.4. 



Adding 15 per cent for cycle imperfections, the compressor will require 
about 66 horsepower. The steam engine required to drive the compressor 
should develop, say, 80 horsepower. 

The volume of the compressor cylinder is, from (8), 

n=§Mi4Mi = i.24c„.ft. 

2 X 75 

Adding 15 per cent for clearance, etc., the required volume is 1.43 cu. ft. 
This is given by a stroke of 20 in. and a cylinder diameter of 12^ in. 



TABLES 



315 



TABLE I 

Pkoperties of Saturated Steam 



§§ 




Heat Content 


Latent Heat 




Entropy 






Ss 


Temp. 
Fahr. 
















Volume 


la 
















OF one 


£si 




of Liquid 


of Vapor 


Total 


Internal 


of Liquid 


of Vapor- 
ization 


of Vapor 


Pound 


V 


t 


./ 


i" 


r 


P 


s' 


r 
T 


&" 


V 


0.5 


58.81 


26.9 


1087.1 


1060.2 


1002.9 


.0532 


2.0431 


2.0963 


1259.3 


1.0 


79.12 


47.2 


1096.7 


1049.5 


989.8 


.0916 


1.9482 


2.0398 


656.7 


1.5 


91.90 


59.9 


1102.5 


1042.6 


982.2 


.1150 


1.8905 


2.0055 


443.0 


2.0 


101.27 


69.2 


1106.6 


1037.4 


975.9 


.1317 


1.8497 


1.9814 


338.3 


2.5 


108.81 


76.7 


1109.9 


1033.2 


970.8 


.1451 


1.8178 


1.9629 


274.3 


3.0 


115.15 


83.1 


1112.7 


1029.6 


966.5 


.1561 


1.7915 


1.9476 


231.2 


3.5 


120.63 


88.5 


1115.0 


1026.5 


962.8 


.1656 


1.7692 


1.9348 


200.1 


4.0 


125.48 


93.4 


1117.1 


1023.7 


959.5 


.1739 


1.7497 


1.9236 


176.6 


4.5 


129.85 


97.7 


1118.9 


1021.2 


956.5 


.1813 


1.7325 


1.9138 


158.1 


5.0 


133.81 


101.7 


1120.6 


1018.9 


953.7 


.1880 


1.7170 


1.9050 


143.2 


6 


140.83 


108.7 


1123.4 


1014.7 


948.8 


.1997 


1.6901 


1.8898 


120.7 


7 


146.90 


114.8 


1125.9 


1011.1 


944.7 


.2097 


1.6672 


1.8769 


104.4 


8 


152.28 


120.2 


1128.0 


1007.9 


940.9 


.2186 


1.6473 


1.8659 


92.2 


9 


157.12 


125.0 


1130.0 


1005.0 


937.5 


.2265 


1.6296 


1.8561 


82.6 


10 


161.52 


129.4 


1131.7 


1002.3 


934.3 


.2336 


1.6138 


1.8474 


74.8 


11 


165.57 


133.4 


1133.3 


999.9 


931.5 


.2401 


1.5994 


1.8395 


68.38 


12 


169.31 


137.2 


1134.7 


997.6 


928.8 


.2460 


1.5862 


1.8322 


63.03 


13 


172.80 


140.6 


1136.0 


995.4 


926.3 


.2516 


1.5739 


1.8254 


58.48 


14 


176.07 


143.9 


1137.3 


993.4 


924.0 


.2568 


1.5627 


1.8195 


54.55 


15 


179.16 


147.0 


1138.5 


991.5 


921.8 


.2616 


1.5522 


1.8138 


51.13 


16 


182.08 


150.0 


1139.6 


989.6 


919.6 


.2662 


1.5423 


1.8085 


48.11 


17 


184.84 


152.7 


1140.6 


987.9 


917.6 


.2705 


1.5330 


1.8035 


45.46 


18 


187.47 


155.4 


1141.6 


986.2 


915.7 


.2746 


1.5242 


1.7988 


43.09 


19 


189.99 


157.9 


1142.5 


984.6 


913.8 


.2785 


1.5158 


1.7943 


40.96 


20 


192.38 


160.3 


1143.4 


983.1 


912.1 


.2822 


1.5079 


1.7901 


39.04 


21 


194.69 


162.6 


1144.2 


981.6 


910.4 


.2857 


1.5003 


1.7860 


37.29 


22 


196.91 


164.8 


1145.0 


980.2 


908.8 


.2891 


1.4931 


1.7822 


35.68 


23 


199.04 


167.0 


1145.8 


978.8 


907.2 


.2923 


1.4862 


1.7785 


34.22 


24 


201.10 


169.0 


1146.5 


977.5 


905.7 


.2955 


1.4796 


1.7751 


32.88 


25 


203.09 


171.0 


1147.2 


976.2 


904.2 


.2985 


1.4732 


1.7717 


31.65 


26 


205.01 


173.0 


1147.9 


974.9 


902.7 


.3014 


1.4670 


1.7674 


30.52 


27 


206.87 


174.9 


1148.6 


973.7 


901.4 


.3042 


1.4611 


1.7653 


29.46 


28 


208.68 


176.7 


1149.2 


972.5 


900.0 


.3069 


1.4554 


1.7623 


28.47 


29 


210.43 


178.4 


1149.8 


971.4 


898.8 


.3095 


1.4499 


1.7594 


27.55 



316 



THERMODYNAMICS 







Heat Content 


Latent Heat 




Entropy 




Pressure 


Temp. 
Fahr. 

t 














Volume 


Lb. per 
Sq. In. 

V 


of Liquid 


of Vapor 


Total 
r 


Internal 
P 


of Liquid 

s' 


Vapori- 
zation 

r 

T 


of Vapor 
s" 


OF One 
Pound 

v" 


14.7 


212.0 


180.0 


1150.4 


970.4 


897.7 


.3121 


1.4449 


1.7570 


26.73 


15 


213.0 


181.1 


1150.8 


969.7 


896.8 


.3137 


1.4417 


1.7554 


26.23 


16 


216.3 


184.4 


1152.0 


967.6 


894.5 


.3186 


1.4315 


1.7501 


24.68 


17 


219.4 


187.5 


1153.0 


965.5 


892.1 


.3231 


1.4220 


1.7451 


23.32 


18 


222.4 


190.5 


1154.1 


963.6 


889.9 


.3275 


1.4129 


1.7404 


22.10 


19 


225.2 


193.4 


1155.0 


961.6 


887.7 


.3317 


1.4043 


1.7360 


21.00 


20 


227.9 


196.1 


1155.9 


959.8 


885.7 


.3357 


1.3961 


1.7318 


20.01 


21 


230.5 


198.7 


1156.8 


958.1 


883.8 


.3395 


1.3883 


1.7278 


19.12 


22 


233.0 


201.2 


1157.7 


956.5 


882.0 


.3431 


1.3809 


1.7240 


18.31 


23 


235.4 


203.6 


1158.5 


954.8 


880.0 


.3467 


1.3738 


1.7205 


17.57 


24 


237.8 


206.0 


1159.2 


953.2 


878.0 


.3501 


1.3669 


1.7170 


16.89 


25 


240.0 


208.3 


1160.0 


951.7 


876.2 


.3533 


1.3604 


1.7137 


16.27 


26 


242.2 


210.5 


1160.7 


950.2 


894.5 


.3565 


1.3540 


1.7105 


15.70 


27 


244.3 


212.7 


1161.5 


948.8 


892.9 


.3595 


1.3479 


1.7074 


15.17 


28 


246.4 


214.8 


1162.1 


947.3 


891.2 


.3625 


1.3419 


1.7044 


14.70 


29 


248.4 


216.8 


1162.8 


946.0 


889.8 


.3653 


1.3362 


1.7015 


14.21 


30 


250.3 


218.8 


1163.4 


944.6 


888.2 


.3681 


1.3307 


1.6988 


13.76 


31 


252.2 


220.7 


1164.0 


943.3 


886.8 


.3708 


1.3253 


1.6961 


13.34 


32 


254.0 


222.5 


1164.6 


942.1 


865.4 


.3734 


1.3201 


1.6935 


12.94 


33 


255.8 


224.3 


1165.1 


940.8 


864.0 


.3760 


1.3151 


1.6911 


12.57 


34 


257.6 


226.1 


1165.7 


939.6 


862.7 


.3784 


1.3102 


1.6886 


12.22 


35 


259.3 


227.8 


1166.2 


938.4 


861.3 


.3808 


1.3054 


1.6863 


11.89 


36 


261.0 


229.5 


1166.8 


937.3 


860.1 


.3832 


1.3008 


1.6840 


11.58 


37 


262.6 


231.2 


1167.3 


936.1 


858.8 


.3855 


1.2963 


1.6818 


11.29 


38 


264.2 


232.8 


1167.8 


935.0 


857.6 


.3878 


1.2918 


1.6796 


11.02 


39 


265.7 


234.4 


1168.3 


933.9 


856.4 


.3900 


1.2876 


1.6776 


10.76 


40 


267.3 


236.0 


1168.8 


932.8 


855.1 


.3921 


1.2834 


1.6755 


10.50 


41 


268.8 


237.5 


1169.3 


931.8 


854.0 


.3942 


1.2793 


1.6735 


10.26 


42 


270.2 


239.0 


1169.7 


930.7 


852.8 


.3962 


1.2753 


1.6715 


10.03 


43 


271.7 


240.5 


1170.1 


929.7 


851.7 


.3982 


1.2714 


1.6696 


9.81 


44 


273.1 


241.9 


1170.6 


928.7 


850.6 


.4002 


1.2676 


1.6678 


9.60 


45 


274.5 


243.3 


1171.0 


927.7 


849.5 


.4021 


1.2638 


1.6659 


9.40 


46 


275.8 


244.6 


1171.4 


926.8 


848.5 


.4040 


1.2602 


1.6642 


9.21 


47 


277.2 


246.0 


1171.8 


925.8 


847.4 


.4059 


1.2566 


1.6625 


9.02 


48 


278.5 


247.3 


1172.2 


924.9 


846.4 


.4077 


1.2531 


1.6608 


8.85 


49 


279.8 


248.7 


1172.6 


923.9 


845.3 


.4095 


1.2496 


1.6591 


8.68 


50 


281.1 


250.0 


1173.0 


923.0 


844.4 


.4112 


1.2463 


1.6575 


8.51 


51 


282.3 


251.3 


1173.4 


922.1 


843.4 


.4130 


1.2429 


1.6559 


8.36 


52 


283.5 


252.6 


1173.8 


921.2 


842.4 


.4147 


1.2397 


1.6544 


8.20 


53 


284.8 


253.8 


1174.2 


920.4 


841.5 


.4164 


1.2365 


1.6529 


8.06 


54 


286.0 


255.0 


1174.5 


919.5 


840.5 


.4180 


1.2333 


1.6513 


7.92 


55 


287.1 


256.2 


1174.9 


918.7 


839.6 


.4196 


1.2303 


1.6499 


7.78 


56 


288.3 


257.4 


1175.2 


917.8 


838.7 


.4212 


1.2272 


1.6484 


7.65 


57 


289.4 


258.6 


1175.6 


917.0 


837.8 


.4228 


1.2243 


1.6471 


7.52 


58 


290.6 


259.7 


1175.9 


916.2 


836.9 


.4243 


1.2213 


1.6456 


7.40 


59 


291.7 


260.8 


1176.2 


915.4 


836.0 


.4258 


1.2184 


1.6442 


7.28 



TABLES 



317 







Heat Content 


Latent Heat 




Entropy 




Pressure 


Temp. 
Fahr. 

t 














Lb. per 
Sq. In. 

P 


of Liquid 
i' 


of Vapor 
i" 


Total 
r 


Internal 
P 


of Liquid 

s' 


Vapori- 
zation 

r 
T 


of Vapor 

s" 


60 


292.8 


262.0 


1176.6 


914.6 


835.2 


.4273 


1.2156 


1.6429 


61 


293.9 


263.1 


1176.9 


913.8 


834.3 


.4288 


1.2128 


1.6416 


62 


294.9 


264.2 


1177.2 


913.0 


833.4 


.4302 


1.2101 


1.6403 


63 


296.0 


265.3 


1177.5 


912.2 


832.5 


.4316 


1.2074 


1.6390 


64 


297.0 


266.3 


1177.8 


911.5 


831.8 


.4330 


1.2047 


1.6378 


65 


298.0 


267.4 


1178.1 


910.7 


830.9 


.4344 


1.2021 


1.6365 


66 


299.1 


268.4 


1178.4 


910.0 


830.2 


.4358 


1.1995 


1.6353 


67 


300.1 


269.5 


1178.7 


909.2 


829.3 


.4372 


1.1969 


1.6341 


68 


301.1 


270.5 


1179.0 


908.5 


828.5 


.4385 


1.1944 


1.6329 


69 


302.0 


271.5 


1179.3 


907.8 


827.7 


.4398 


1.1920 


1.6318 


70 


303.0 


272.4 


1179.5 


907.1 


827.0 


.4411 


1.1895 


1.6306 


71 


303.9 


273.4 


1179.8 


906.4 


826.2 


.4424 


1.1871 


1.6295 


72 


304.9 


274.4 


1180.1 


905.7 


825.5 


.4437 


1.1847 


1.6284 


73 


305.8 


275.4 


1180.4 


905.0 


824.7 


.4450 


1.1823 


1.6273 


74 


306.8 


276.3 


1180.6 


904.3 


823.9 


.4462 


1.1800 


1.6262 


75 


307.7 


277.3 


1180.9 


903.6 


823.2 


.4474 


1.1777 


1.6251 


76 


308.6 


278.2 


1181.1 


902.9 


822.4 


.4486 


1.1755 


1.6241 


77 


309.5 


279.1 


1181.4 


902.3 


821.8 


.4498 


1.1732 


1.6230 


78 


310.4 


280.0 


1181.6 


901.6 


821.0 


.4510 


1.1710 


1.6220 


79 


311.2 


280.9 


1181.9 


901.0 


820.4 


.4522 


1.1688 


1.6210 


80 


312.1 


281.8 


1182.1 


900.3 


819.6 


.4533 


1.1667 


1.6200 


82 


313.8 


283.6 


1182.7 


899.0 


818.2 


.4556 


1.1625 


1.6181 


84 


315.5 


285.3 


1183.1 


897.8 


816.9 


.4578 


1.1584 


1.6162 


86 


317.2 


287.0 


1183.6 


896.6 


815.6 


.4600 


1.1543 


1.6143 


88 


318.8 


288.7 


1184.0 


895.4 


814.3 


.4622 


1.1503 


1.6125 


90 


320.4 


290.3 


1184.5 


894.2 


813.0 


.4642 


1.1465 


1.6107 


92 


321.9 


291.9 


1184.9 


893.0 


811.7 


.4663 


1.1427 


1.6090 


94 


323.4 


293.5 


1185.3 


891.8 


810.4 


.4683 


1.1390 


1.6073 


96 


324.9 


295.0 


1185.7 


890.7 


809.3 


.4703 


1.1353 


1.6056 


98 


326.5 


296.6 


1186.1 


889.5 


808.0 


.4723 


1.1317 


1.6040 


100 


327.9 


298.1 


1186.5 


888.4 


806.8 


.4742 


1.1282 


1.6024 


102 


329.3 


299.6 


1186.9 


887.3 


805.6 


.4761 


1.1248 


1.6009 


104 


330.7 


301.0 


1187.3 


886.3 


804.6 


.4779 


1.1214 


1.5993 


106 


332.1 


302.5 


1187.7 


885.2 


803.4 


.4797 


1.1181 


1.5978 


108 


333.5 


303.9 


1188.1 


884.2 


802.3 


.4815 


1.1149 


1.5964 


110 


334.8 


305.3 


1188.4 


883.1 


801.1 


.4833 


1.1117 


1.5950 


112 


336.2 


306.7 


1188.8 


882.1 


800.0 


.4850 


1.1085 


1.5935 


114 


337.5 


308.0 


1189.1 


881.1 


799.0 


.4867 


1.1054 


1.5921 


116 


338.8 


309.4 


1189.5 


880.1 


797.9 


.4884 


1.1024 


1.5908 


118 


340.1 


310.7 


1189.8 


879.1 


796.8 


.4901 


1.0994 


1.5895 


120 


341.3 


312.0 


1190.1 


878.2 


795.8 


.4917 


1.0965 


1.5882 


122 


342.6 


313.3 


1190.5 


877.2 


794.8 


.4933 


1.0936 


1.5869 


124 


343.8 


314.5 


1190.8 


876.3 


793.8 


.4949 


1.0907 


1.5856 


126 


345.0 


315.8 


1191.1 


875.3 


792.7 


.4965 


1.0879 


1.5844 


128 


346.2 


317.0 


1191.4 


874.4 


791.8 


.4980 


1.0851 


1.5831 



Volume 
OF One 
Pound 



7.168 
7.057 
6.949 
6.845 
6.744 

6.646 
6.551 
6.459 
6.370 
6.283 

6.198 
6.115 
6.035 
5.957 

5.882 

5.809 
5.737 
5.666 
5.597 
5.530 

5.464 
5.338 
5.219 
5.104 
4.995 

4.890 
4.789 
4.692 
4.599 
4.511 

4.425 
4.343 
4.264 

4.188 
4.114 

4.043 
3.975 
3.909 
3.845 

3.784 

3.724 
3.666 
3.610 
3.555 
3.502 



318 



THERMODYNAMICS 



Pressure 
Lb. per 
Sq. In. 



130 
132 
134 
136 
138 

140 
142 
144 
146 
148 

150 
160 
170 
180 
190 

200 
210 
220 
230 
240 

250 
260 
270 
280 
290 

300 



Temp. 
Fahr. 


Heat Content 


Latent Heat 




Entropy 




of Liquid 


of Vapor 


Total 


Internal 


of Liquid 


Vapori- 
zation 


of Vapor 


t 


l' 


^" 


r 


P 


s' 


r 
T 


s" 


347.4 


318.2 


1191.7 


873.5 


790.8 


.4995 


1.0824 


1.5819 


348.6 


319.4 


1192.0 


872.6 


789.9 


.5010 


1.0797 


1.5807 


349.7 


320.6 


1192.3 


871.7 


788.9 


.5025 


1.0770 


1.5795 


350.8 


321.8 


1192.6 


870.8 


788.0 


.5039 


1.0744 


1.5783 


352.0 


323.0 


1192.9 


869.9 


787.0 


.5054 


1.0719 


1.5773 


353.1 


324.2 


1193.2 


869.0 


786.1 


.5068 


1.0693 


1.5761 


354.2 


325.3 


1193.5 


868.2 


785.2 


.5082 


1.0668 


1.5750 


355.3 


326.5 


1193.8 


867.3 


784.3 


.5096 


1.0644 


1.5740 


356.4 


327.6 


1194.0 


866.5 


783.4 


.5110 


1.0619 


1.5729 


357.4 


328.7 


1194.3 


865.6 


782.4 


.5123 


1.0595 


1.5718 


358.5 


329.8 


1194.6 


864.8 


781.6 


.5137 


1.0571 


1.5708 


363.6 


335.0 


1195.8 


860.8 


777.4 


.5202 


1.0456 


1.5658 


368.5 


340.2 


1197.1 


856.9 


773.2 


.5263 


1.0349 


1.5612 


373.1 


345.0 


1198.2 


853.2 


769.3 


.5321 


1.0246 


1.5567 


377.6 


349.6 


1199.3 


849.6 


765.5 


.5377 


1.0149 


1.5526 


381.8 


354.1 


1200.3 


846.2 


762.0 


.5430 


1.0057 


1.5487 


385.9 


358.4 


1201.3 


842.9 


758.5 


.5481 


.9968 


1.5449 


389.9 


362.5 


1202.2 


839.6 


755.1 


.5530 


.9884 


1.5414 


393.7 


366.5 


1203.0 


836.5 


751.8 


.5577 


.9803 


1.5380 


397.4 


370.4 


1203.9 


833.5 


748.7 


.5622 


.9726 


1.5348 


401.0 


374.1 


1204.7 


830.6 


745.7 


.5666 


.9651 


1.5317 


404.5 


377.8 


1205.5 


827.7 


742.7 


.5708 


.9579 


1.5287 


407.8 


381.3 


1206.2 


824.9 


739.8 


.5749 


.9510 


1.5259 


411.1 


384.7 


1206.9 


822.2 


737.0 


.5788 


.9443 


1.5231 


414.3 


388.1 


1207.6 


819.5 


734.2 


.5826 


.9378 


1.5204 


417.4 


391.3 


1208.3 


817.0 


731.5 


.5863 


.9315 


1.5178 



Volume 
OP One 
Pound 



3.452 
■3.402 
3.354 
3.307 
3.262 

3.218 
3.175 
3.133 
3.092 
3.052 

3.014 

2.834 
2.676 
2.534 
2.407 

2.292 
2.188 
2.093 
2.006 
1.926 

1.852 
1.784 
1.722 
1.663 
1.608 

1.558 



TABLES 



319 



TABLE II 



Properties of Saturated Steam below 212° F. 





Pressure 




Weight of One Cubic 






J 


Temp. 






Volume of 

One Pound 

(Cu. Ft.) 


r OC 




Total 
Heat 


Latent 
Heat 


Temp. 


Fahr. 


Lb. per 
Sq. Iq. 


Inches 
of Hg. 


Pounds 


Grains 


Fahr. 


t 


P 


- 


v" 


7 


— 


q" 


r 


t 


32 


0.0885 


0.1802 


3288 


0.000304 


2.129 


1073.7 


1073.7 


32 


34 


0.0960 


0.1955 


3047 


0.000328 


2.297 


1074.7 


1072.7 


34 


36 


0.1039 


0.2116 


2826 


0.000354 


2.477 


1075.8 


1071.7 


36 


38 


0.1125 


0.2291 


2623 


0.000381 


2.669 


1076.8 


1070.8 


38 


40 


0.1217 


0.2478 


2437 


0.000410 


2.872 


1077.8 


1069.8 


40 


42 


0.1315 


0.2677 


2265 


0.000442 


3.091 


1078.8 


1068.8 


42 


44 


0.1420 


0.2891 


2107 


0.000475 


3.322 


1079.8 


1067.7 


44 


46 


0.1532 


0.3119 


1961 


0.000510 


3.570 


1080.8 


1066.7 


46 


48 


0.1653 


0.3366 


1827 


0.000547 


3.831 


1081.8 


1065.7 


48 


50 


0.1781 


0.3627 


1703 


0.000587 


4.110 


1082.8 


1064.7 


50 


52 


0.1918 


0.3905 


1587 


0.000630 


4.411 


1083.8 


1063.7 


52 


54 


0.2064 


0.4202 


1483 


0.000674 


4.720 


1084.7 


1062.7 


54 


56 


0.2219 


0.4518 


1385 


0.000722 


5.054 


1085.7 


1061.6 


56 


58 


0.2385 


0.4856 


1294 


0.000773 


5.410 


1086.7 


1060.6 


58 


60 


0.2562 


0.5217 


1210 


0.000827 


5.785 


1087.6 


1059.6 


60 


62 


0.2749 


0.5598 


1132 


0.000883 


6.043 


1088.6 


1058.5 


62 


64 


0.2949 


0.6005 


1060 


0.000943 


6.604 


1089.6 


1057.5 


64 


66 


0.3161 


0.644 


993 


0.001007 


7.048 


1090.5 


1056.4 


66 


68 


0.3386 


0.689 


931 


0.001074 


7.52 


1091.5 


1055.4 


68 


70 


0.3625 


0.738 


873 


0.001145 


8.02 


1092.4 


1054.3 


70 


72 


0.3879 


0.789 


820 


0.001220 


8.54 


1093.3 


1053.3 


72 


74 


0.4148 


0.844 


770 


0.001300 


9.10 


1094.3 


1052.2 


74 


76 


0.4433 


0.903 


723 


0.001383 


9.68 


1095.2 


1051.2 


76 


78 


0.4735 


0.964 


680 


0.001471 


10.30 


1096.1 


1050.1 


78 


80 


0.5054 


1.029 


639 


0.001564 


10.95 


1097.1 


1049.0 


80 


82 


0.539 


1.098 


601.4 


0.001663 


11.64 


1098.0 


1048.0 


82 


84 


0.575 


1.171 


565.7 


0.001768 


12.37 


1098.9 


1046.9 


84 


86 


0.613 


1.248 


532.2 


0.001879 


13.15 


1099.8 


1045.8 


86 


88 


0.653 


1.329 


500.8 


0.001997 


13.98 


1100.7 


1044.7 


88 


90 


0.695 


1.415 


471.4 


0.002121 


14.85 


1101.6 


1043.6 


90 


92 


0.739 


1.506 


443.9 


0.002253 


15.77 


1102.5 


1042.5 


92 


94 


0.787 


1.602 


418.2 


0.002391 


16.74 


1103.4 


1041.4 


94 


96 


0.837 


1.704 


394.2 


0.002537 


17.76 


1104.3 


1040.3 


96 


98 


0.890 


1.812 


371.8 


0.002690 


18.79 


1105.2 


1039.2 


98 


100 


0.946 


1.925 


350.9 


0.002580 


19.95 


1106.1 


1038.1 


100 



320 



THERMODYNAMICS 



Temp 


Pressure 


Volume of 

One Pound 

(Cu. Ft.) 


Weight of One Cubic 
Foot 


Total 
Heat 


Latent 
Heat 


Temp 


Fahr. 


Lb. per 
Sq. In. 


Inches of 
Hg. 


Pounds 


Grains 


Fahr. 


t 


V 


— 


v" 


7 


— 


q" 


r 


t 


102 
104 
106 
108 
110 


1.004 
1.066 
1.131 
1.199 
1.271 


2.044 
2.171 
2.303 
2.441 

2.588 


331.4 
313.2 
296.2 
280.4 
265.6 


0.003017 
0.003193 
0.003376 
0.003566 
0.003765 


21.12 
22.35 
23.63 
24.96 
26.36 


1107.0 
1107.9 
1108.7 
1109.6 
1110.5 


1037.0 
1035.9 
1034.8 
1033.7 
1032.5 


102 
104 
106 
108 
110 


120 
130 
140 
150 
160 


1.689 
2.219 
2.885 
3.714 
4.737 


3.439 
4.518 

5.874 

7.56 

9.64 


203.4 

157.5 

123.1 

97.2 

77.4 


0.004916 

0.00635 

0.00812 

0.01029 

0.01293 


34.42 

44.45 

56.86 

72.0 

90.5 


1114.8 
1119.0 
1123.1 
1127.1 
1131.1 


1026.9 
1021.1 
1015.2 
1009.3 
1003.2 


120 
130 
140 
150 
160 


170 
180 
190 
200 
210 


5.988 

7.506 

9.335 

11.523 

14.122 


12.19 
15.28 
90.01 
23.46 

28.75 


. 62.09 
50.23 
40.94 
33.60 

27.77 


0.01611 
0.01991 
0.02443 
0.02976 
0.03601 


112.7 
139.4 
171.0 
208.3 
252.1 


1135.0 
1138.8 
1142.5 
1146.2 
1149.7 


997.1 
990.9 
984.6 
978.2 
971.7 


170 
180 
190 
200 
210 


212 


14.697 


29.92 


26.75 


0.03738 


261.7 


1150.4 


970.4 


212 



TABLES 



321 



o 



o 
I— I f> 



Q 

fa 



< 



o 

g 

o 



Tempera- 
ture, F. 


OlOOkOOlOOlOOlOOlOOIOOlOOlOOlOOlOOiOOiOO 
CO<N(NrHrH 1 i-H tH Cfl N CO CO -"it Tt» « »0 «0 «0 t- t- 00 00 0> O O 

1 1 1 1 1 ^ 


Vt>LUME OF 

One Pound 

v" 


GO^COCOOOCO^:OOOCOOi'-HT-HOO^r-^coO'-^Tt^l-lOrHTf^OOTt^,-^0 

t^coc^^'NtHoqiococoTt^^coo^oO'*ococooi>iocoi-HOii>coLOTj^ 
a5^>^l6co^doio6t>^o:dlOT^Tl^^o6c6coc^c^'(^^(N'-^T-^T--^r-^^ 


{H 

a, 
o 

1 


Vaporization 
r 
T 


Ot>>O(MO00»OC0T-(0iC0'stH(MO00<:O'*(MO00CD-^(Mi-i05l>iO 
^C0C0C0C0(NCq(M(MT-iT-H^^^OOOOO0505C3i05C5000000 




cocorHa5oocoTtHcqT-io5i:^coioi>ooo5rH(McoTj<Locoi>oooooiO 

-*CO(NOaiOOt>COiOCO(NrHOOrH(NTt^i001>00050rHCqcOiO 
^^^^OOOOOOOOOOOOOOOOOO^^rHrHT^ 

1 1 1 i 1 1 1 1 1 1 1 1 1 




■3 

1 - 

X! 




% <a. 


^05iOO<:0(NI>(NI>COOOCOOOC005COOOCOOOCOOO(Mt^C^OOiO 
»O'*TtH'*C0C0(M(M'-HT-HOO0i0i0000I>l><:OC0iOiOTtHTtHC0(r0CQ 


of Vapori- 
zation 

r 


coC5LO'--it>cooO"^'*5:itoo<:OrHi>c^i>c^i>c^t>c<ii:0'--icDOrt<ai 


1 




col>ooOl-^(^^coT}^locD^>oooiO'-H(^^(^^(^OTt^Ti^loto^:ol>l>l>oo 


1 


t>Oqt^^CO^iOO"*a5COOO(NC005iOOCO(Ml>COC5iO'-H>COC55 
OCOiOiO'^^COCOtNrH^ 1 1 rH (M (M CO CO 'st^ ^ »0 CD CO l> l> 

1 1 1 1 1 1 1 1 1 1 1 ' ' 


Pressure 
Lb. per 
Sq. In. 


i-Hrt^lMCOOOlMGOOqO^COOOOOa 

ioiooocO'-Hcoi>iooort^'^aio^cococDiOi-Hiocoiooqoq(N(>q 
col0^^dc6ddco^-^c^'^>^(^idLd(^qdo6^>^^>^^>^odd(^idr-^dco 

'-H^rH<M(M(N(NCOCOTjHTtliO>OCOt>0000050'-H(M'*»OC00005i-H 

rHi-HT-HT-Hr-li-Hi-HT-HC^ 




« 


0100100»00»00»00»00»OOIOOIOOU30100»00»00 
COC4C4r-l«H i tH tH C4 C4 CO CO ^ ^ lO lO CO CO t> t« 00 00 O) O) O 

Mill' "^ 



INDEX 



[The numbers refer to pages] 



Absolute scale, Kelvin's, 55. 

temperature, 18. 

zero, 18. 
Acoustic velocity, 257. 
Adiabatic change, defined, 40. 

expansion of gas, 103. 

of vapor mixture, 185, 189. 
of superheated steam, 218. 

irreversible, 75. 

of air and steam mixture, 233. 

of superheated steam, approximation 
to, 220. 

of vapor mixture, approximation to, 
190. 

on r*S-plane, 70. 

with variable specific heat, 126. 
Air and steam, mixture of, 232, 236. 

compression, 152. 

engine cycles, analysis of, 140. 

engines, classification of, 137. 

moist, constants for, 230. 

moisture in, 228. 

refrigeration, 149. 

required for combustion, 119. 
Allen dense-air refrigerating machine, 

150. 
Ammonia, saturated, 180. 

superheated, 223. 
Andrews' experiments, 198. 
Atomic weights. 111. 
Availability of energy, 46. 
Available energy of a system, 56. 

Bertrand's formulas, 168. 

Biot's formula, 167. 

Boltzmann's interpretation of the second 

law, 65. 
Boyle's law, 89. 
Brayton cycle, 145. 

Callendar's equation for superheated 

steam, 204. 
Calorimeter, throttling, 271. 
Caloric theory, 3. 
Carbon dioxide, saturated, 182. 
Carnot cycle, 50, 134. 

for saturated vapors, 283. 

on TS-plane, 73. 

engine, efficiency of, 54. 



Carnot's principle, 52. 
Characteristic equation, 16. 

of gases, 93, 277. 

surface, 20. 
Charles' law, 90. 
Chemical energy, 5. 
Clapeyron-Clausius formula, 178. 
Clausius' equation, 200. 

inequality of, 63. 

statement of the second law, 50. 
Combustion, 117. 

air required for, 119. 

products of, 119. 

temperature of, 127. 
Compound compression of air, 156. 
Compounding of steam turbines, 296. 
Compressed air, 152. 

engines, 158. 
Compression, compound, 156. 

refrigerating machine, 308. 
Conduction of heat, waste in, 57. 
Conservation of energy, 6. 
Constant energy curve of mixture, 187. 
Constant volume curve, 186. 
Continuity, equation of, 244. 
Coordinates defining state of system, 15. 
Critical states, 197. 

temperature, volume, and pressure, 199. 
Cycle, Carnot, 50, 134. 

Diesel, 146. 

Joule, 145. 

Lenoir, 162. 

Otto, 142. 

processes, 72, 133. 

Rankine, 284. 

rectangular, 73. 
Cycles, isoadiabatic, 136. 

of actual steam engine, 290. 

of air engines, analysis of, 140. 

of gas engines, comparison of, 148. 

with irreversible adiabatics, 75. 
Cylinder efficiency, 293. 
Curtis type of steam turbine, 304. 
Curve, constant volume, of steam, 186. 

of heating and cooling, 70. 

polytropic, 71. 

saturation, 166, 182. 
Curves, specific heat, superheated steam, 
209, 211. 



323 



324 



INDEX 



Dalton'slaw, 114, 228. 
Davis formula for heat content, 177, 274. 
Degradation of energy, 7. 
Degree of superheat, 165, 196. 
De Laval nozzle, 258. 
Derivative ^' 170. 
Design of nozzles, 264. 
Diesel cycle, 146. 

Differential equations of thermodynam- 
ics, 82, 84. 

expressions, interpretation of, 28. 

inexact, 30. 
Differentials oi u, i, F and *, 79 . 
Dissociation, 197. 
Dupre-Hertz formula, 168. 

Efficiency, conditions of maximum, 135. 

cyHnder, 293. 

of Carnot engine, 54. 

potential, 292. 

ratio, 292. 

thermal, 291. 

standards, 291. 
Electrical energy, 5. 
Energy, availability of, 46. 

chemical, 5. 

conservation of, 6. 

degradation of, 7. 

dissipation of, 8. 

electrical, 5. 
Energy equation, 36. 

applied to cycle process, 39. 
applied to vaporization, 170. 
integration of, 38. 
Energy, heat, 3. 

high grade, and low grade, 7. 

mechanical, 2. 

of gases, 97. 

of saturated vapor, 172. 

of superheated steam, 214. 

relativity of, 2. 

transformations of, 5. 

units of, 8. 

units, relations between, 10. 
Engine, compressed air, 158. 

Ericsson's, 139. 

Stirhng's, 138. 
Engines, gas, 142. 

hot-air, 138. 

steam, 283. 
Entropy, as a co5rdinate, 68. 

first definition of, 59. 

of gases, 100. 

of hquid, 179. 

of superheated steam, 215. 

of vapor, 179. 

principle, application of, 60. 

second definition of, 62. 
Equation of continuity, 244. 



Equation of Clausius, 200. 

of perfect gas, 17. 

of van der Waals, 20, 200. 

of vapor mixture, 184. 
Equations for gases, 94. 

for discharge of air and steam, 255. 

for superheated steam, 203. 

general, of thermodynamics, 79. 
Equilibrium of thermodynamics systems, 

87. 
Ericsson's air engine, 139. 
Exact differentials, 30. 
Expansion of gases, adiabatic, 103. 

at constant pressure, 101. 

isothermal, 102. 
Expansion valve, 272, 309. 
Exponent n, determination of, 108. 
External work of a system, 37. 

First law of thermodynamics, 35. 
Fliegner's equations for flow of air, 255. 
Flow of air, equations for, 255. 
Flow of fluids, assumptions, 244. 

experiments on, 243, 254. 

formulas for discharge, 255 

fundamental equations, 244. 

graphical representation, 247. 
through orifices, 252. 
Flow of steam, Grashof's equation, 256. 

Rateau's equation, 256. 

Napier's equation, 257. 
Free expansion of gases, 58. 
Friction in nozzles, 262. 
Frictional processes, 74, 
Fuels, 118. 

Gas, characteristic equation of, 93, 277. 

constant B, value of, 92. 

constant, universal, 113. 

constants, relations between, 112. 

free expansion of, 58. 

permanent, 89. 
Gas-engine cycles, comparison of, 148. 
Gases, entropy of, 100. 

general equations for, 94. 

heat content of, 99. 

intrinsic energy of, 97. 

laws of , 89. 

mixtures of, 114. 

specific heat of, 96, 124. 
Graphical representation of energy equa- 
tion, 43. 

of flow of fluids, 247. 
Grashof's equation, flow of steam, 256. 

Heat absorbed during change of state, 22. 
conduction, waste in, 57. 
content, Davis formula for, 177, 274. 
content, defined, 77. 



INDEX 



325 



Heat content of gases, 99. 

of saturated vapor, 173, 177. 
of superheated steam, 210. 
Heat, effects of, 35. 

latent, 26. 

mechanical equivalent of, 11. 

mechanical theory of, 3. 

of liquid, 171, 174. 

of vaporization, 171, 175. 

specific, 24. 

total, 172, 177, 213. 

units of, 9. 
Heating of air by internal combustion, 

141. 
Heating value of fuels, 118. 
Henning's formula for latent heat, 176. 
Holborn and Henning's experiments, 

205. 
Hot-air engines, 138. 
Humidity, 229. 

Inequality of Clausius, 63. 

Internal combustion, heating by, 141. 

Intrinsic energy, 36. 

of gases, 97. 

of superheated steam, 214. 
of vapors, 172. 
Irreversible adiabatics, 75. 
processes, 47. 
processes, waste in, 57. 
Isoadiabatic cycles, 136. 
Isodynamic change of vapor, 190. 

processes, 42. 
Isometric lines, 22. 
Isopiestic lines, 22. 
Isothermal, definition of, 21. 
expansion of gases, 102. 
of superheated steam, 217. 
of vapor mixture, 188. 
on TS-plsine, 70. 
of steam and air mixture, 232. ' 

Jet, work of, 298. 
Joule's cycle, 145. 

experiments, 11. 

law, 90. 
Joule-Thomson coefficient, 276. 
effect, 275. 

Kelvin's absolute scale, 55. 

statement of the second law, 50. 
Knoblauch's experiments, 201. 
Knoblauch and Jakob's experiments, 205. 

Langen's equations for specific heat, 124, 

205. 
Latent heat, 26. 

external, 172. 

Henning's formula for, 176. 



Latent heat, internal, 172. 

of expansion, 27. 

of pressure variation, 27. 

of vaporization, 171, 175. 
Laws of gases, 89. 
Lenoir cycle, 162. 

Linde's process for liquefaction, 280. 
Liquefaction of gases, 280. 
Liquid curve, 166. 

Mallard and Le Chatelier's experiments, 

205. 
Marks' formula, 170. 
Maxwell's thermodynamic relations, 80 
Mean specific heat, 210. 
Mechanical energy, units of, 9. 
Mechanical equivalent of heat, 11. 

theory of heat, 3. 
Mixture of gases and vapors, 228. 

of gases, specific heat of, 125. 

of steam and air, 232, 236. 
Moist air," constants for, 230. 
Moisture in atmosphere, 228. 
Molecular specific heat, 123. 

weights. 111. 
Mollier's chart, 223. 

use in flow of fluids, 251. 
use in steam turbines, 302. 
Munich experiments, 201. 

Napier's equations, flow of steam, 257. 
Nozzle, De Laval, 258. 
Nozzles, design of, 264. 
friction in, 262. 

Otto cycle, 142, 148. 

Peake's throttling curves, 273. 
Perfect gas, definition of, 18. 

equation of, 17. 
Permanent gas, explanation of term, 

89. 
Perpetual motion of first class, 6. 

of second class, 8. 
Polytropic change of state, 104. 

changes, specific heat in, 106. 

curve, 71. 
Potential efficiency, 292. 

thermodynamic, 77, 87. 
Pressure and temperature, relation be- 
tween, 167. 
Pressure compounding,. 296. 

critical, 199. 

turbines, action of, 298, 305. 
Products of combustion, 119. 

Quality of mixture, 165. 
variation of, 185. 



326 



INDEX 



Rankine's cycle, 284. 

effect of changing pressure, 

289. 
incomplete expansion, 288. 
with superheated steam, 
286. 
formula, 168. 
Rateau's formula, flow of steam, 286. 
Rectangular cycle, 73. 
Refrigerating machine, analysis of, 311. 
Refrigeration, air, 149. 
vapors used in, 310. 
with vapor media, 308. 
Reversed heat engine, 74. 
Reversible processes, 47. 
Reynolds and Moorby's experiments, 12. 
Rowland's experiments, 11. 
Rotary engine, 294. 

Saint Venant's hypothesis, 254. 
Saturated vapor, 165. 

energy of, 172. 
entropy of, 179. 
heat content of, 173, 177. 
latent heat of, 171, 175. 
specific heat of, 182. 
surface representing, 166. 
total heat of, 172, 177. 
Saturation curve, 166, 182. 

temperature, 165. 
Second law of thermodynamics, 50. 

Boltzmann's interpretation of, 65. 
Specific heat, 24. 

curves, 209, 211. 
in poly tropic changes, 106. 
Langen's formulas for, 124. 
mean, 210. 
Specific heat, molecular, 123. 

of gaseous mixture, 125. 
of gaseous products, 123. 
of gases, 96. 

of saturated vapor, 182. 
of superheated steam, 204, 273. 
Specific volume of vapors, 177. 
Steam and air, mixture of, 232, 236. 
•critical temperature of, 199. 
specific volume of, 177. 
tables, 180. 

thermal properties of, 173. 
total heat of, 172, 177. 
Steam turbine, 294. 

classification of, 295. 
compared with reciprocating 

engine 294, 
compounding, 296. 
Curtis type, 304. 
impulse and reaction, 296. 
influence of high vacuum, 307. 
low pressure, 307. 



Steam turbine multiple stage, 302. 
pressure type, 298, 305. 
single stage, 300. 
velocity and pressure, 296. 
Stirling's engine, 138. 
Sulphur dioxide, saturated, 182. 

superheated, 223. 
Superheat, degree of, 165, 196. 
Superheated ammonia, 223. 
Superheated steam, 165, 196. 

changes of state, 216. 
energy of, 214. 
entropy of, 215. 
equations for, 203. 
heat content of, 210. 
specific heat of, 204, 273. 
tables and diagrams, 221. 
total heat of, 213. 
Superheated sulphur dioxide, 223. 
• vapor, characteristics of, 196. 
Surface, characteristic, 20. 

representing saturated vapor, 166 
System, defined, 15. 
state of, 15. 

Temperature, absolute, 18. 

and pressure, relation between, 167. 

critical, 199. 

Kelvin scale of, 55. 
Temperature of combustion, 127. 

saturation, 165. 

scales, comparison of, 91. 
Temperature entropy representation, 68. 
Thermal capacities, relation between, 27. 

capacity defined, 24. 

efficiency, 291. 

energy, 4. 

lines, 21. 

properties of steam, 173. 
Thermodynamic degeneration, 8. 

potentials, 77, 87. 

relations, 80. 
Thermodynamics, first law of, 35. 

general equations of, 84. 

scope of, 1. 

second law of, 50. 
Throttling calorimeter, 271. 

curves, 273. 

loss due to, 269. 

processes, 268. 
Total heat of saturated vapor, 172, 177. 

of superheated steam, 213. 
Transformations of energy, 5. 
Tumlirtz equation for superheated steam, 

204. 
Turbine, steam, see Steam turbine. 

Units of energy, 8. 

of heat, 9. 
Universal gas constant, 113. 



INDEX 



327 



Vacuum, influence of, on steam turbine, 

307. 
Van der Waals' equation, 20, 200. 
Vapor, energy of, 172. 

entropy of, 179. 

heat content of, 173, 177. 

latent heat of, 171, 175. 
Vapor mixture, adiabatic expansion of, 
189. 

constant volume change, 189. 

curves on T*S-plane, 186. 

general equation of, 184. 

isodynamic of, 190. 

isothermal expansion of, 188. 
Vapor refrigerating machine, 311. 

superheated, 196. 

total heat of, 172, 177. 
Vaporization, heat of, 171, 175. 



Vaporization, process of, 164. 
Vapors used in refrigeration, 310. 
Velocity compounding, 297. 
Volume, critical, 199. 
specific, of vapor, 177. 

Waste in irreversible processes, 57. 
Water, critical temperature of, 199. 

jacketing, 155. 

vapor, thermal properties of, 173. 
Wiredrawing, 268. 
Work, conversion of, into heat, 57. 

external, of expansion, 37. 

of a jet, 298. 

Zero curve, 186. 

Zeuner's equation for superheated steam, 
204. 



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